1
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Swailem M, Täuber UC. Computing macroscopic reaction rates in reaction-diffusion systems using Monte Carlo simulations. Phys Rev E 2024; 110:014124. [PMID: 39160995 DOI: 10.1103/physreve.110.014124] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/08/2024] [Accepted: 06/21/2024] [Indexed: 08/21/2024]
Abstract
Stochastic reaction-diffusion models are employed to represent many complex physical, biological, societal, and ecological systems. The macroscopic reaction rates describing the large-scale, long-time kinetics in such systems are effective, scale-dependent renormalized parameters that need to be either measured experimentally or computed by means of a microscopic model. In a Monte Carlo simulation of stochastic reaction-diffusion systems, microscopic probabilities for specific events to happen serve as the input control parameters. To match the results of any computer simulation to observations or experiments carried out on the macroscale, a mapping is required between the microscopic probabilities that define the Monte Carlo algorithm and the macroscopic reaction rates that are experimentally measured. Finding the functional dependence of emergent macroscopic rates on the microscopic probabilities (subject to specific rules of interaction) is a very difficult problem, and there is currently no systematic, accurate analytical way to achieve this goal. Therefore, we introduce a straightforward numerical method of using lattice Monte Carlo simulations to evaluate the macroscopic reaction rates by directly obtaining the count statistics of how many events occur per simulation time step. Our technique is first tested on well-understood fundamental examples, namely, restricted birth processes, diffusion-limited two-particle coagulation, and two-species pair annihilation kinetics. Next we utilize the thus gained experience to investigate how the microscopic algorithmic probabilities become coarse-grained into effective macroscopic rates in more complex model systems such as the Lotka-Volterra model for predator-prey competition and coexistence, as well as the rock-paper-scissors or cyclic Lotka-Volterra model and its May-Leonard variant that capture population dynamics with cyclic dominance motifs. Thereby we achieve a more thorough and deeper understanding of coarse graining in spatially extended stochastic reaction-diffusion systems and the nontrivial relationships between the associated microscopic and macroscopic model parameters, with a focus on ecological systems. The proposed technique should generally provide a useful means to better fit Monte Carlo simulation results to experimental or observational data.
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Affiliation(s)
- Mohamed Swailem
- Department of Physics & Center for Soft Matter and Biological Physics, MC 0435, Robeson Hall, 850 West Campus Drive, Virginia Tech, Blacksburg, Virginia 24061, USA
| | - Uwe C Täuber
- Department of Physics & Center for Soft Matter and Biological Physics, MC 0435, Robeson Hall, 850 West Campus Drive, Virginia Tech, Blacksburg, Virginia 24061, USA
- Faculty of Health Sciences, Virginia Tech, Blacksburg, Virginia 24061, USA
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2
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Nag Chowdhury S, Banerjee J, Perc M, Ghosh D. Eco-evolutionary cyclic dominance among predators, prey, and parasites. J Theor Biol 2023; 564:111446. [PMID: 36868345 DOI: 10.1016/j.jtbi.2023.111446] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/04/2022] [Revised: 02/24/2023] [Accepted: 02/25/2023] [Indexed: 03/05/2023]
Abstract
Predator-prey interactions are one of ecology's central research themes, but with many interdisciplinary implications across the social and natural sciences. Here we consider an often-overlooked species in these interactions, namely parasites. We first show that a simple predator-prey-parasite model, inspired by the classical Lotka-Volterra equations, fails to produce a stable coexistence of all three species, thus failing to provide a biologically realistic outcome. To improve this, we introduce free space as a relevant eco-evolutionary component in a new mathematical model that uses a game-theoretical payoff matrix to describe a more realistic setup. We then show that the consideration of free space stabilizes the dynamics by means of cyclic dominance that emerges between the three species. We determine the parameter regions of coexistence as well as the types of bifurcations leading to it by means of analytical derivations as well as by means of numerical simulations. We conclude that the consideration of free space as a finite resource reveals the limits of biodiversity in predator-prey-parasite interactions, and it may also help us in the determination of factors that promote a healthy biota.
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Affiliation(s)
- Sayantan Nag Chowdhury
- Department of Environmental Science and Policy, University of California, Davis, CA 95616, USA
| | - Jeet Banerjee
- BYJU'S, Think & Learn Pvt. Ltd., IBC Knowledge Park, 4/1 Bannerghatta Main Road, Bangalore 560029, India
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia; Alma Mater Europaea, Slovenska ulica, 17, 2000 Maribor, Slovenia; Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan; Complexity Science Hub Vienna, Josefstädterstraße 39, 1080 Vienna, Austria; Department of Physics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul, Republic of Korea
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700108, India.
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3
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Universal scaling of extinction time in stochastic evolutionary dynamics. Sci Rep 2022; 12:22403. [PMID: 36575301 PMCID: PMC9794815 DOI: 10.1038/s41598-022-27102-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/18/2022] [Accepted: 12/26/2022] [Indexed: 12/28/2022] Open
Abstract
Evolutionary dynamics is well captured by the replicator equations when the population is infinite and well-mixed. However, the extinction dynamics is modified with finite and structured populations. Experiments on the non-transitive ecosystem containing three populations of bacteria found that the ecological stability sensitively depends on the spatial structure of the populations. Based on the Reference-Gamble-Birth algorithm, we use agent-based Monte Carlo simulations to investigate the extinction dynamics in the rock-paper-scissors ecosystem with finite and structured populations. On the fully-connected network, the extinction time in stable and unstable regimes falls into two universal functions when plotted with the rescaled variables. On the two dimensional grid, the spatial structure changes the transition boundary between stable and unstable regimes but doesn't change its extinction trend. The finding of universal scaling in extinction dynamics is unexpected, and may provide a powerful method to classify different evolutionary dynamics into universal classes.
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4
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Mir H, Stidham J, Pleimling M. Emerging spatiotemporal patterns in cyclic predator-prey systems with habitats. Phys Rev E 2022; 105:054401. [PMID: 35706181 DOI: 10.1103/physreve.105.054401] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/20/2022] [Accepted: 04/15/2022] [Indexed: 06/15/2023]
Abstract
Three-species cyclic predator-prey systems are known to establish spiral waves that allow species to coexist. In this study, we analyze a structured heterogeneous system which gives one species an advantage to escape predation in an area that we refer to as a habitat and study the effect on species coexistence and emerging spatiotemporal patterns. Counterintuitively, the predator of the advantaged species emerges as dominant species with the highest average density inside the habitat. The species given the advantage in the form of an escape rate has the lowest average density until some threshold value for the escape rate is exceeded, after which the density of the species with the advantage overtakes that of its prey. Numerical analysis of the spatial density of each species as well as of the spatial two-point correlation function for both inside and outside the habitats allow a detailed quantitative discussion. Our analysis is extended to a six-species game that exhibits spontaneous spiral waves, which displays similar but more complicated results.
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Affiliation(s)
- Hana Mir
- Department of Physics and Center for Soft Matter and Biological Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA
| | - James Stidham
- Department of Physics and Center for Soft Matter and Biological Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA
| | - Michel Pleimling
- Department of Physics and Center for Soft Matter and Biological Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA
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5
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Avelino PP, de Oliveira BF, Trintin RS. Lotka-Volterra versus May-Leonard formulations of the spatial stochastic rock-paper-scissors model: The missing link. Phys Rev E 2022; 105:024309. [PMID: 35291086 DOI: 10.1103/physreve.105.024309] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/06/2021] [Accepted: 02/04/2022] [Indexed: 06/14/2023]
Abstract
The rock-paper-scissors (RPS) model successfully reproduces some of the main features of simple cyclic predator-prey systems with interspecific competition observed in nature. Still, lattice-based simulations of the spatial stochastic RPS model are known to give rise to significantly different results, depending on whether the three-state Lotka-Volterra or the four-state May-Leonard formulation is employed. This is true independently of the values of the model parameters and of the use of either a von Neumann or a Moore neighborhood. In this paper, we introduce a simple modification to the standard spatial stochastic RPS model in which the range of the search of the nearest neighbor may be extended up to a maximum Euclidean radius R. We show that, with this adjustment, the Lotka-Volterra and May-Leonard formulations can be designed to produce similar results, both in terms of dynamical properties and spatial features, by means of an appropriate parameter choice. In particular, we show that this modified spatial stochastic RPS model naturally leads to the emergence of spiral patterns in both its three- and four-state formulations.
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Affiliation(s)
- P P Avelino
- Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal
- Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, PT4169-007 Porto, Portugal
| | - B F de Oliveira
- Departamento de Física, Universidade Estadual de Maringá, Av. Colombo 5790, 87020-900 Maringá, PR, Brazil
| | - R S Trintin
- Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal
- Departamento de Física, Universidade Estadual de Maringá, Av. Colombo 5790, 87020-900 Maringá, PR, Brazil
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6
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Islam S, Mondal A, Mobilia M, Bhattacharyya S, Hens C. Effect of mobility in the rock-paper-scissor dynamics with high mortality. Phys Rev E 2022; 105:014215. [PMID: 35193192 DOI: 10.1103/physreve.105.014215] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/27/2021] [Accepted: 01/11/2022] [Indexed: 06/14/2023]
Abstract
In the evolutionary dynamics of a rock-paper-scissor model, the effect of natural death plays a major role in determining the fate of the system. Coexistence, being an unstable fixed point of the model, becomes very sensitive toward this parameter. In order to study the effect of mobility in such a system which has explicit dependence on mortality, we perform Monte Carlo simulation on a two-dimensional lattice having three cyclically competing species. The spatiotemporal dynamics has been studied along with the two-site correlation function. Spatial distribution exhibits emergence of spiral patterns in the presence of mobility. It reveals that the joint effect of death rate and mobility (diffusion) leads to new coexistence and extinction scenarios.
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Affiliation(s)
- Sahil Islam
- Department of Physics, Jadavpur University, Jadavpur, Kolkata 700032, India
| | - Argha Mondal
- Department of Mathematics, Sidho-Kanho-Birsha University, Purulia 723104, WB, India
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom
| | - Mauro Mobilia
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
| | | | - Chittaranjan Hens
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
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7
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8
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Czuppon P, Traulsen A. Understanding evolutionary and ecological dynamics using a continuum limit. Ecol Evol 2021; 11:5857-5873. [PMID: 34141189 PMCID: PMC8207364 DOI: 10.1002/ece3.7205] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/06/2020] [Revised: 12/08/2020] [Accepted: 12/23/2020] [Indexed: 01/08/2023] Open
Abstract
Continuum limits in the form of stochastic differential equations are typically used in theoretical population genetics to account for genetic drift or more generally, inherent randomness of the model. In evolutionary game theory and theoretical ecology, however, this method is used less frequently to study demographic stochasticity. Here, we review the use of continuum limits in ecology and evolution. Starting with an individual-based model, we derive a large population size limit, a (stochastic) differential equation which is called continuum limit. By example of the Wright-Fisher diffusion, we outline how to compute the stationary distribution, the fixation probability of a certain type, and the mean extinction time using the continuum limit. In the context of the logistic growth equation, we approximate the quasi-stationary distribution in a finite population.
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Affiliation(s)
- Peter Czuppon
- Institute of Ecology and Environmental Sciences ParisUPECCNRSIRDINRASorbonne UniversitéParisFrance
- Centre Interdisciplinaire de Recherche en BiologieCNRSCollège de FrancePSL Research UniversityParisFrance
- Department of Evolutionary TheoryMax Planck Institute for Evolutionary BiologyPlönGermany
| | - Arne Traulsen
- Department of Evolutionary TheoryMax Planck Institute for Evolutionary BiologyPlönGermany
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9
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Ursell T. Structured environments foster competitor coexistence by manipulating interspecies interfaces. PLoS Comput Biol 2021; 17:e1007762. [PMID: 33412560 PMCID: PMC7790539 DOI: 10.1371/journal.pcbi.1007762] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/05/2020] [Accepted: 10/19/2020] [Indexed: 01/12/2023] Open
Abstract
Natural environments, like soils or the mammalian gut, frequently contain microbial consortia competing within a niche, wherein many species contain genetically encoded mechanisms of interspecies competition. Recent computational work suggests that physical structures in the environment can stabilize local competition between species that would otherwise be subject to competitive exclusion under isotropic conditions. Here we employ Lotka-Volterra models to show that interfacial competition localizes to physical structures, stabilizing competitive ecological networks of many species, even with significant differences in the strength of competitive interactions between species. Within a limited range of parameter space, we show that for stable communities the length-scale of physical structure inversely correlates with the width of the distribution of competitive fitness, such that physical environments with finer structure can sustain a broader spectrum of interspecific competition. These results highlight the potentially stabilizing effects of physical structure on microbial communities and lay groundwork for engineering structures that stabilize and/or select for diverse communities of ecological, medical, or industrial utility.
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Affiliation(s)
- Tristan Ursell
- Institute of Molecular Biology, University of Oregon, Eugene, Oregon, United States of America
- Materials Science Institute, University of Oregon, Eugene, Oregon, United States of America
- Department of Physics, University of Oregon, Eugene, Oregon, United States of America
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10
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Bhattacharyya S, Sinha P, De R, Hens C. Mortality makes coexistence vulnerable in evolutionary game of rock-paper-scissors. Phys Rev E 2020; 102:012220. [PMID: 32795013 DOI: 10.1103/physreve.102.012220] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/09/2020] [Accepted: 07/13/2020] [Indexed: 11/07/2022]
Abstract
Multiple species in the ecosystem are believed to compete cyclically for maintaining balance in nature. The evolutionary dynamics of cyclic interaction crucially depends on different interactions representing different natural habits. Based on a rock-paper-scissors model of cyclic competition, we explore the role of mortality of individual organisms in the collective survival of a species. For this purpose a parameter called "natural death" is introduced. It is meant for bringing about the decease of an individual irrespective of any intra- and interspecific interaction. We perform a Monte Carlo simulation followed by a stability analysis of different fixed points of defined rate equations and observe that the natural death rate is surprisingly one of the most significant factors in deciding whether an ecosystem would come up with a coexistence or a single-species survival.
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Affiliation(s)
| | - Pritam Sinha
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
| | - Rina De
- Department of Physics, R.R.R Mahavidyalaya, Radhanagar, Hooghly 712406, India
| | - Chittaranjan Hens
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
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11
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Baker R, Pleimling M. The effect of habitats and fitness on species coexistence in systems with cyclic dominance. J Theor Biol 2020; 486:110084. [PMID: 31758965 DOI: 10.1016/j.jtbi.2019.110084] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/30/2019] [Accepted: 11/15/2019] [Indexed: 11/25/2022]
Abstract
Cyclic dominance between species may yield spiral waves that are known to provide a mechanism enabling persistent species coexistence. This observation holds true even in presence of spatial heterogeneity in the form of quenched disorder. In this work we study the effects on spatio-temporal patterns and species coexistence of structured spatial heterogeneity in the form of habitats that locally provide one of the species with an advantage. Performing extensive numerical simulations of systems with three and six species we show that these structured habitats destabilize spiral waves. Analyzing extinction events, we find that species extinction probabilities display a succession of maxima as function of time, that indicate a periodically enhanced probability for species extinction. Analysis of the mean extinction time reveals that as a function of the parameter governing the advantage of one of the species a transition between stable coexistence and unstable coexistence takes place. We also investigate how efficiency as a predator or a prey affects species coexistence.
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Affiliation(s)
- Ryan Baker
- Academy of Integrated Science, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0563, USA
| | - Michel Pleimling
- Academy of Integrated Science, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0563, USA; Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0435, USA; Center for Soft Matter and Biological Physics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0435, USA.
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12
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Cremer J, Melbinger A, Wienand K, Henriquez T, Jung H, Frey E. Cooperation in Microbial Populations: Theory and Experimental Model Systems. J Mol Biol 2019; 431:4599-4644. [PMID: 31634468 DOI: 10.1016/j.jmb.2019.09.023] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/03/2019] [Revised: 09/25/2019] [Accepted: 09/26/2019] [Indexed: 01/07/2023]
Abstract
Cooperative behavior, the costly provision of benefits to others, is common across all domains of life. This review article discusses cooperative behavior in the microbial world, mediated by the exchange of extracellular products called public goods. We focus on model species for which the production of a public good and the related growth disadvantage for the producing cells are well described. To unveil the biological and ecological factors promoting the emergence and stability of cooperative traits we take an interdisciplinary perspective and review insights gained from both mathematical models and well-controlled experimental model systems. Ecologically, we include crucial aspects of the microbial life cycle into our analysis and particularly consider population structures where ensembles of local communities (subpopulations) continuously emerge, grow, and disappear again. Biologically, we explicitly consider the synthesis and regulation of public good production. The discussion of the theoretical approaches includes general evolutionary concepts, population dynamics, and evolutionary game theory. As a specific but generic biological example, we consider populations of Pseudomonas putida and its regulation and use of pyoverdines, iron scavenging molecules, as public goods. The review closes with an overview on cooperation in spatially extended systems and also provides a critical assessment of the insights gained from the experimental and theoretical studies discussed. Current challenges and important new research opportunities are discussed, including the biochemical regulation of public goods, more realistic ecological scenarios resembling native environments, cell-to-cell signaling, and multispecies communities.
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Affiliation(s)
- J Cremer
- Department of Molecular Immunology and Microbiology, Groningen Biomolecular Sciences and Biotechnology Institute, University of Groningen, 9747 AG Groningen, the Netherlands
| | - A Melbinger
- Arnold-Sommerfeld-Center for Theoretical Physics and Center for Nanoscience, Ludwig-Maximilians-Universität München, Theresienstrasse 37, D-80333 Munich, Germany
| | - K Wienand
- Arnold-Sommerfeld-Center for Theoretical Physics and Center for Nanoscience, Ludwig-Maximilians-Universität München, Theresienstrasse 37, D-80333 Munich, Germany
| | - T Henriquez
- Microbiology, Department of Biology I, Ludwig-Maximilians-Universität München, Grosshaderner Strasse 2-4, Martinsried, Germany
| | - H Jung
- Microbiology, Department of Biology I, Ludwig-Maximilians-Universität München, Grosshaderner Strasse 2-4, Martinsried, Germany.
| | - E Frey
- Arnold-Sommerfeld-Center for Theoretical Physics and Center for Nanoscience, Ludwig-Maximilians-Universität München, Theresienstrasse 37, D-80333 Munich, Germany.
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13
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Brown BL, Meyer-Ortmanns H, Pleimling M. Dynamically generated hierarchies in games of competition. Phys Rev E 2019; 99:062116. [PMID: 31330747 DOI: 10.1103/physreve.99.062116] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/24/2018] [Indexed: 02/05/2023]
Abstract
Spatial many-species predator-prey systems have been shown to yield very rich space-time patterns. This observation begs the question whether there exist universal mechanisms for generating this type of emerging complex patterns in nonequilibrium systems. In this work we investigate the possibility of dynamically generated hierarchies in predator-prey systems. We analyze a nine-species model with competing interactions and show that the studied situation results in the spontaneous formation of spirals within spirals. The parameter dependence of these intriguing nested spirals is elucidated. This is achieved through the numerical investigation of various quantities (correlation lengths, densities of empty sites, Fourier analysis of species densities, interface fluctuations) that allows us to gain a rather complete understanding of the spatial arrangements and the temporal evolution of the system. A possible generalization of the interaction scheme yielding dynamically generated hierarchies is discussed. As cyclic interactions occur spontaneously in systems with competing strategies, the mechanism discussed in this work should contribute to our understanding of various social and biological systems.
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Affiliation(s)
- Barton L Brown
- Department of Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA.,Center for Soft Matter and Biological Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA
| | | | - Michel Pleimling
- Department of Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA.,Center for Soft Matter and Biological Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA.,Academy of Integrated Science, Virginia Tech, Blacksburg, Virginia 24061-0563, USA
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14
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Bassler KE, Frey E, Zia RKP. Coevolution of nodes and links: Diversity-driven coexistence in cyclic competition of three species. Phys Rev E 2019; 99:022309. [PMID: 30934283 DOI: 10.1103/physreve.99.022309] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/20/2018] [Indexed: 06/09/2023]
Abstract
When three species compete cyclically in a well-mixed, stochastic system of N individuals, extinction is known to typically occur at times scaling as the system size N. This happens, for example, in rock-paper-scissors games or conserved Lotka-Volterra models in which every pair of individuals can interact on a complete graph. Here we show that if the competing individuals also have a "social temperament" to be either introverted or extroverted, leading them to cut or add links, respectively, then long-living states in which all species coexist can occur. These nonequilibrium quasisteady states only occur when both introverts and extroverts are present, thus showing that diversity can lead to stability in complex systems. In this case, it enables a subtle balance between species competition and network dynamics to be maintained.
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Affiliation(s)
- Kevin E Bassler
- Department of Physics, University of Houston, Houston, Texas 77204-5005, USA; Texas Center for Superconductivity, University of Houston, Houston, Texas 77204-5002, USA; and Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Strasse 38, Dresden D-01187, Germany
| | - Erwin Frey
- Arnold Sommerfeld Center for Theoretical Physics and Center for Nanoscience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333 München, Germany
| | - R K P Zia
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Strasse 38, Dresden D-01187, Germany and Center for Soft Matter and Biological Physics, Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA
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15
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Nagatani T, Ichinose G, Tainaka KI. Metapopulation dynamics in the rock-paper-scissors game with mutation: Effects of time-varying migration paths. J Theor Biol 2019; 462:425-431. [DOI: 10.1016/j.jtbi.2018.11.030] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/03/2018] [Revised: 11/24/2018] [Accepted: 11/27/2018] [Indexed: 12/19/2022]
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16
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Peruani F, Sibona GJ. Reaction processes among self-propelled particles. SOFT MATTER 2019; 15:497-503. [PMID: 30601543 DOI: 10.1039/c8sm01502c] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
We study a system of self-propelled disks that perform run-and-tumble motion, where particles can adopt more than one internal state. One of those internal states can be transmitted to other particle if the particle carrying this state maintains physical contact with another particle for a finite period of time. We refer to this process as a reaction process and to the different internal states as particle species, making an analogy to chemical reactions. The studied system may fall into an absorbing phase, where due to the disappearance of one of the particle species no further reaction can occur, or may remain in an active phase where particles constantly react. By combining individual-based simulations and mean-field arguments, we study the dependency of the equilibrium densities of particle species on motility parameters, specifically the active speed v0 and tumbling frequency λ. We find that the equilibrium densities of particle species exhibit two very distinct, non-trivial scaling regimes, with v0 and λ depending on whether the system is in the so-called ballistic or diffusive regime. Our mean-field estimates lead to an effective renormalization of reaction rates that allow building the phase-diagram v0-λ that separates the absorbing and active phases. We find an excellent agreement between numerical simulations and mean-field estimates. This study is a necessary step towards an understanding of phase transitions into absorbing states in active systems and sheds light on the spreading of information/signaling among moving elements.
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Affiliation(s)
- Fernando Peruani
- Université Côte d'Azur, Laboratoire J.A. Dieudonné, UMR CNRS 7351, Parc Valrose, F-06108 Nice Cedex 02, France.
| | - Gustavo J Sibona
- CONICET and Fa.M.A.F., Universidad Nacional de Córdoba, Córdoba, Argentina.
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17
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Metapopulation model of rock-scissors-paper game with subpopulation-specific victory rates stabilized by heterogeneity. J Theor Biol 2018; 458:103-110. [PMID: 30213665 DOI: 10.1016/j.jtbi.2018.09.009] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2018] [Revised: 09/07/2018] [Accepted: 09/10/2018] [Indexed: 11/20/2022]
Abstract
Recently, metapopulation models for rock-paper-scissors games have been presented. Each subpopulation is represented by a node on a graph. An individual is either rock (R), scissors (S) or paper (P); it randomly migrates among subpopulations. In the present paper, we assume victory rates differ in different subpopulations. To investigate the dynamic state of each subpopulation (node), we numerically obtain the solutions of reaction-diffusion equations on the graphs with two and three nodes. In the case of homogeneous victory rates, we find each subpopulation has a periodic solution with neutral stability. However, when victory rates between subpopulations are heterogeneous, the solution approaches stable focuses. The heterogeneity of victory rates promotes the coexistence of species.
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18
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Metapopulation model for rock-paper-scissors game: Mutation affects paradoxical impacts. J Theor Biol 2018; 450:22-29. [PMID: 29627264 DOI: 10.1016/j.jtbi.2018.04.005] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2018] [Revised: 04/01/2018] [Accepted: 04/03/2018] [Indexed: 11/20/2022]
Abstract
The rock-paper-scissors (RPS) game is known as one of the simplest cyclic dominance models. This game is key to understanding biodiversity. Three species, rock (R), paper (P) and scissors (S), can coexist in nature. In the present paper, we first present a metapopulation model for RPS game with mutation. Only mutation from R to S is allowed. The total population consists of spatially separated patches, and the mutation occurs in particular patches. We present reaction-diffusion equations which have two terms: reaction and migration terms. The former represents the RPS game with mutation, while the latter corresponds to random walk. The basic equations are solved analytically and numerically. It is found that the mutation induces one of three phases: the stable coexistence of three species, the stable phase of two species, and a single-species phase. The phase transitions among three phases occur by varying the mutation rate. We find the conditions for coexistence are largely changed depending on metapopulation models. We also find that the mutation induces different paradoxes in different patches.
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19
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Heterogeneous network promotes species coexistence: metapopulation model for rock-paper-scissors game. Sci Rep 2018; 8:7094. [PMID: 29728573 PMCID: PMC5935761 DOI: 10.1038/s41598-018-25353-4] [Citation(s) in RCA: 26] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/08/2018] [Accepted: 04/19/2018] [Indexed: 11/08/2022] Open
Abstract
Understanding mechanisms of biodiversity has been a central question in ecology. The coexistence of three species in rock-paper-scissors (RPS) systems are discussed by many authors; however, the relation between coexistence and network structure is rarely discussed. Here we present a metapopulation model for RPS game. The total population is assumed to consist of three subpopulations (nodes). Each individual migrates by random walk; the destination of migration is randomly determined. From reaction-migration equations, we obtain the population dynamics. It is found that the dynamic highly depends on network structures. When a network is homogeneous, the dynamics are neutrally stable: each node has a periodic solution, and the oscillations synchronize in all nodes. However, when a network is heterogeneous, the dynamics approach stable focus and all nodes reach equilibriums with different densities. Hence, the heterogeneity of the network promotes biodiversity.
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20
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West R, Mobilia M, Rucklidge AM. Survival behavior in the cyclic Lotka-Volterra model with a randomly switching reaction rate. Phys Rev E 2018; 97:022406. [PMID: 29548111 DOI: 10.1103/physreve.97.022406] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/23/2017] [Indexed: 11/07/2022]
Abstract
We study the influence of a randomly switching reproduction-predation rate on the survival behavior of the nonspatial cyclic Lotka-Volterra model, also known as the zero-sum rock-paper-scissors game, used to metaphorically describe the cyclic competition between three species. In large and finite populations, demographic fluctuations (internal noise) drive two species to extinction in a finite time, while the species with the smallest reproduction-predation rate is the most likely to be the surviving one (law of the weakest). Here we model environmental (external) noise by assuming that the reproduction-predation rate of the strongest species (the fastest to reproduce and predate) in a given static environment randomly switches between two values corresponding to more and less favorable external conditions. We study the joint effect of environmental and demographic noise on the species survival probabilities and on the mean extinction time. In particular, we investigate whether the survival probabilities follow the law of the weakest and analyze their dependence on the external noise intensity and switching rate. Remarkably, when, on average, there is a finite number of switches prior to extinction, the survival probability of the predator of the species whose reaction rate switches typically varies nonmonotonically with the external noise intensity (with optimal survival about a critical noise strength). We also outline the relationship with the case where all reaction rates switch on markedly different time scales.
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Affiliation(s)
- Robert West
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
| | - Mauro Mobilia
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
| | - Alastair M Rucklidge
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
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21
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Emergence of unusual coexistence states in cyclic game systems. Sci Rep 2017; 7:7465. [PMID: 28785001 PMCID: PMC5547111 DOI: 10.1038/s41598-017-07911-4] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/04/2017] [Accepted: 07/04/2017] [Indexed: 11/08/2022] Open
Abstract
Evolutionary games of cyclic competitions have been extensively studied to gain insights into one of the most fundamental phenomena in nature: biodiversity that seems to be excluded by the principle of natural selection. The Rock-Paper-Scissors (RPS) game of three species and its extensions [e.g., the Rock-Paper-Scissors-Lizard-Spock (RPSLS) game] are paradigmatic models in this field. In all previous studies, the intrinsic symmetry associated with cyclic competitions imposes a limitation on the resulting coexistence states, leading to only selective types of such states. We investigate the effect of nonuniform intraspecific competitions on coexistence and find that a wider spectrum of coexistence states can emerge and persist. This surprising finding is substantiated using three classes of cyclic game models through stability analysis, Monte Carlo simulations and continuous spatiotemporal dynamical evolution from partial differential equations. Our finding indicates that intraspecific competitions or alternative symmetry-breaking mechanisms can promote biodiversity to a broader extent than previously thought.
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22
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Souza-Filho CA, Bazeia D, Ramos JGGS. Apex predator and the cyclic competition in a rock-paper-scissors game of three species. Phys Rev E 2017; 95:062411. [PMID: 28709300 DOI: 10.1103/physreve.95.062411] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2017] [Indexed: 06/07/2023]
Abstract
This work deals with the effects of an apex predator on the cyclic competition among three distinct species that follow the rules of the rock-paper-scissors game. The investigation develops standard stochastic simulations but is motivated by a procedure which is explained in the work. We add the apex predator as the fourth species in a system that contains three species that evolve following the standard rules of migration, reproduction, and predation, and study how the system evolves in this new environment, in comparison with the case in the absence of the apex predator. The results show that the apex predator engenders the tendency to spread uniformly in the lattice, contributing to destroy the spiral patterns, keeping biodiversity but diminishing the average size of the clusters of the species that compete cyclically.
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Affiliation(s)
- C A Souza-Filho
- Departamento de Física, Universidade Federal da Paraíba, 58051-970 João Pessoa, Paraíba, Brazil
- Instituto Federal de Educação, Ciência e Tecnologia da Paraíba, Campus Princesa Isabel, 58755-000, Princesa Isabel, Paraíba, Brazil
| | - D Bazeia
- Departamento de Física, Universidade Federal da Paraíba, 58051-970 João Pessoa, Paraíba, Brazil
| | - J G G S Ramos
- Departamento de Física, Universidade Federal da Paraíba, 58051-970 João Pessoa, Paraíba, Brazil
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23
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Feldager CW, Mitarai N, Ohta H. Deterministic extinction by mixing in cyclically competing species. Phys Rev E 2017; 95:032318. [PMID: 28415345 DOI: 10.1103/physreve.95.032318] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2016] [Indexed: 11/07/2022]
Abstract
We consider a cyclically competing species model on a ring with global mixing at finite rate, which corresponds to the well-known Lotka-Volterra equation in the limit of infinite mixing rate. Within a perturbation analysis of the model from the infinite mixing rate, we provide analytical evidence that extinction occurs deterministically at sufficiently large but finite values of the mixing rate for any species number N≥3. Further, by focusing on the cases of rather small species numbers, we discuss numerical results concerning the trajectories toward such deterministic extinction, including global bifurcations caused by changing the mixing rate.
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Affiliation(s)
- Cilie W Feldager
- Center for Models of Life, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark
| | - Namiko Mitarai
- Center for Models of Life, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark
| | - Hiroki Ohta
- Niels Bohr International Academy/Center for Models of Life, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark
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24
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Weber MF, Frey E. Master equations and the theory of stochastic path integrals. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2017; 80:046601. [PMID: 28306551 DOI: 10.1088/1361-6633/aa5ae2] [Citation(s) in RCA: 31] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a 'generating functional', which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a 'forward' and a 'backward' path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon expanding the forward and the backward path integrals around stationary paths, we then discuss and extend a recent method for the computation of rare event probabilities. Besides, we also derive path integral representations for processes with continuous state spaces whose forward and backward master equations admit Kramers-Moyal expansions. A truncation of the backward expansion at the level of a diffusion approximation recovers a classic path integral representation of the (backward) Fokker-Planck equation. One can rewrite this path integral in terms of an Onsager-Machlup function and, for purely diffusive Brownian motion, it simplifies to the path integral of Wiener. To make this review accessible to a broad community, we have used the language of probability theory rather than quantum (field) theory and do not assume any knowledge of the latter. The probabilistic structures underpinning various technical concepts, such as coherent states, the Doi-shift, and normal-ordered observables, are thereby made explicit.
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Affiliation(s)
- Markus F Weber
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, 80333 München, Germany
| | - Erwin Frey
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, 80333 München, Germany
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25
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Stewart AJ, Parsons TL, Plotkin JB. Evolutionary consequences of behavioral diversity. Proc Natl Acad Sci U S A 2016; 113:E7003-E7009. [PMID: 27791109 PMCID: PMC5111714 DOI: 10.1073/pnas.1608990113] [Citation(s) in RCA: 34] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Iterated games provide a framework to describe social interactions among groups of individuals. This body of work has focused primarily on individuals who face a simple binary choice, such as "cooperate" or "defect." Real individuals, however, can exhibit behavioral diversity, varying their input to a social interaction both qualitatively and quantitatively. Here we explore how access to a greater diversity of behavioral choices impacts the evolution of social dynamics in populations. We show that, in public goods games, some simple strategies that choose between only two possible actions can resist invasion by all multichoice invaders, even while engaging in relatively little punishment. More generally, access to a larger repertoire of behavioral choices results in a more "rugged" fitness landscape, with populations able to stabilize cooperation at multiple levels of investment. As a result, increased behavioral choice facilitates cooperation when returns on investments are low, but it hinders cooperation when returns on investments are high. Finally, we analyze iterated rock-paper-scissors games, the nontransitive payoff structure of which means that unilateral control is difficult to achieve. Despite this, we find that a large proportion of multichoice strategies can invade and resist invasion by single-choice strategies-so that even well-mixed populations will tend to evolve and maintain behavioral diversity.
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Affiliation(s)
- Alexander J Stewart
- Department of Genetics, Environment and Evolution, University College London, London WC1E 6BT, United Kingdom;
| | - Todd L Parsons
- Laboratoire de Probabilités et Modèles Aléatoires, CNRS UMR 7599, Université Pierre et Marie Curie, Paris 75005, France
| | - Joshua B Plotkin
- Department of Biology, University of Pennsylvania, Philadelphia, PA 19104
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26
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The Influence of Mobility Rate on Spiral Waves in Spatial Rock-Paper-Scissors Games. GAMES 2016. [DOI: 10.3390/g7030024] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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27
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Roman A, Dasgupta D, Pleimling M. A theoretical approach to understand spatial organization in complex ecologies. J Theor Biol 2016; 403:10-16. [DOI: 10.1016/j.jtbi.2016.05.009] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/28/2016] [Accepted: 05/05/2016] [Indexed: 02/02/2023]
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28
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Wang T, Duan J, Liu T. Competition promotes the persistence of populations in ecosystems. Sci Rep 2016; 6:30477. [PMID: 27460143 PMCID: PMC4962045 DOI: 10.1038/srep30477] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/24/2016] [Accepted: 07/05/2016] [Indexed: 11/29/2022] Open
Abstract
Competition is one of the most common form in ecological systems, which plays important roles in population dynamics. However, the influences of competition on persistence of populations remain unclear when space effect is included. In this paper, we investigated a predator-prey model with competition and spatial diffusion. Based on pattern formations and time series of populations, we found that competitions induce the persistence of populations, which denies competitive exclusion principle. Moreover, we testify the robustness of these effects. Our results also suggest that space may lead to the emergence of new phenomenon in ecosystems.
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Affiliation(s)
- Tao Wang
- College of Science, Shihezi University, Shihezi, Xinjiang 832003, P. R. China
| | - Jinqiao Duan
- Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA
- Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, 430074, P. R. China
| | - Tong Liu
- College of Life Science, Shihezi University, Shihezi, Xinjiang 832003, P. R. China
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29
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Brett T, Galla T. Generating functionals and Gaussian approximations for interruptible delay reactions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:042105. [PMID: 26565166 DOI: 10.1103/physreve.92.042105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/19/2015] [Indexed: 06/05/2023]
Abstract
We develop a generating functional description of the dynamics of non-Markovian individual-based systems in which delay reactions can be terminated before completion. This generalizes previous work in which a path-integral approach was applied to dynamics in which delay reactions complete with certainty. We construct a more widely applicable theory, and from it we derive Gaussian approximations of the dynamics, valid in the limit of large, but finite, population sizes. As an application of our theory we study predator-prey models with delay dynamics due to gestation or lag periods to reach the reproductive age. In particular, we focus on the effects of delay on noise-induced cycles.
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Affiliation(s)
- Tobias Brett
- Theoretical Physics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom
- Department of Ecology and Evolutionary Biology, University of Michigan, Ann Arbor, Michigan 48109, USA
- Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48109, USA
| | - Tobias Galla
- Theoretical Physics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom
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30
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Pei QM, Zhan X, Yang LJ, Shen J, Wang LF, Qui K, Liu T, Kirunda JB, Yousif AAM, Li AB, Jia Y. Fluctuation and noise propagation in phenotypic transition cascades of clonal populations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012721. [PMID: 26274216 DOI: 10.1103/physreve.92.012721] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/03/2015] [Indexed: 06/04/2023]
Abstract
Quantitative modeling of fluctuations of each phenotype is a crucial step towards a fundamental understanding of noise propagation through various phenotypic transition cascades. The theoretical formulas for noise propagation in various phenotypic transition cascades are derived by using the linear noise approximation of master equation and the logarithmic gain. By virtue of the theoretical formulas, we study the noise propagation in bidirectional and unidirectional phenotypic transition cascades, respectively. It is found that noise propagation in these two phenotypic transition cascades evidently differs: In the bidirectional cascade, a systemic random environment is provided by a correlated global component. The total noise of each phenotype is mainly determined by the intrinsic noise and the transmitted noise from other phenotypes. The intrinsic noise enlarged by interconversion through an added part shows a novel noise propagation mechanism. However, in the unidirectional cascade, the random environment of each downstream phenotype is provided by upstream phenotypes. The total noise of each downstream phenotype is mainly determined by the transmitted noises from upstream phenotypes. The intrinsic noise and the conversion noise can propagate in both bidirectional and unidirectional phenotypic transition cascades.
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Affiliation(s)
- Qi-ming Pei
- Institute of Biophysics and Department of Physics, Central China Normal University, Wuhan 430079, China
- School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China
| | - Xuan Zhan
- Institute of Biophysics and Department of Physics, Central China Normal University, Wuhan 430079, China
| | - Li-jian Yang
- Institute of Biophysics and Department of Physics, Central China Normal University, Wuhan 430079, China
| | - Jian Shen
- Institute of Biophysics and Department of Physics, Central China Normal University, Wuhan 430079, China
| | - Li-fang Wang
- Institute of Biophysics and Department of Physics, Central China Normal University, Wuhan 430079, China
| | - Kang Qui
- Institute of Biophysics and Department of Physics, Central China Normal University, Wuhan 430079, China
| | - Ting Liu
- Institute of Biophysics and Department of Physics, Central China Normal University, Wuhan 430079, China
| | - J B Kirunda
- Institute of Biophysics and Department of Physics, Central China Normal University, Wuhan 430079, China
| | - A A M Yousif
- Institute of Biophysics and Department of Physics, Central China Normal University, Wuhan 430079, China
| | - An-bang Li
- Institute of Biophysics and Department of Physics, Central China Normal University, Wuhan 430079, China
| | - Ya Jia
- Institute of Biophysics and Department of Physics, Central China Normal University, Wuhan 430079, China
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31
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Intoy B, Pleimling M. Synchronization and extinction in cyclic games with mixed strategies. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:052135. [PMID: 26066147 DOI: 10.1103/physreve.91.052135] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2015] [Indexed: 06/04/2023]
Abstract
We consider cyclic Lotka-Volterra models with three and four strategies where at every interaction agents play a strategy using a time-dependent probability distribution. Agents learn from a loss by reducing the probability to play a losing strategy at the next interaction. For that, an agent is described as an urn containing β balls of three and four types, respectively, where after a loss one of the balls corresponding to the losing strategy is replaced by a ball representing the winning strategy. Using both mean-field rate equations and numerical simulations, we investigate a range of quantities that allows us to characterize the properties of these cyclic models with time-dependent probability distributions. For the three-strategy case in a spatial setting we observe a transition from neutrally stable to stable when changing the level of discretization of the probability distribution. For large values of β, yielding a good approximation to a continuous distribution, spatially synchronized temporal oscillations dominate the system. For the four-strategy game the system is always neutrally stable, but different regimes emerge, depending on the size of the system and the level of discretization.
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Affiliation(s)
- Ben Intoy
- Department of Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA
| | - Michel Pleimling
- Department of Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA
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32
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Grošelj D, Jenko F, Frey E. How turbulence regulates biodiversity in systems with cyclic competition. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:033009. [PMID: 25871204 DOI: 10.1103/physreve.91.033009] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/14/2014] [Indexed: 06/04/2023]
Abstract
Cyclic, nonhierarchical interactions among biological species represent a general mechanism by which ecosystems are able to maintain high levels of biodiversity. However, species coexistence is often possible only in spatially extended systems with a limited range of dispersal, whereas in well-mixed environments models for cyclic competition often lead to a loss of biodiversity. Here we consider the dispersal of biological species in a fluid environment, where mixing is achieved by a combination of advection and diffusion. In particular, we perform a detailed numerical analysis of a model composed of turbulent advection, diffusive transport, and cyclic interactions among biological species in two spatial dimensions and discuss the circumstances under which biodiversity is maintained when external environmental conditions, such as resource supply, are uniform in space. Cyclic interactions are represented by a model with three competitors, resembling the children's game of rock-paper-scissors, whereas the flow field is obtained from a direct numerical simulation of two-dimensional turbulence with hyperviscosity. It is shown that the space-averaged dynamics undergoes bifurcations as the relative strengths of advection and diffusion compared to biological interactions are varied.
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Affiliation(s)
- Daniel Grošelj
- Max-Planck-Institut für Plasmaphysik, Boltzmannstraße 2, D-85748 Garching, Germany
| | - Frank Jenko
- Max-Planck-Institut für Plasmaphysik, Boltzmannstraße 2, D-85748 Garching, Germany
- Department of Physics and Astronomy, University of California, Los Angeles, California 90095-1547, USA
| | - Erwin Frey
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 München, Germany
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33
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Mesoscopic interactions and species coexistence in evolutionary game dynamics of cyclic competitions. Sci Rep 2014; 4:7486. [PMID: 25501627 PMCID: PMC4265771 DOI: 10.1038/srep07486] [Citation(s) in RCA: 54] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2014] [Accepted: 11/27/2014] [Indexed: 11/18/2022] Open
Abstract
Evolutionary dynamical models for cyclic competitions of three species (e.g., rock, paper, and scissors, or RPS) provide a paradigm, at the microscopic level of individual interactions, to address many issues in coexistence and biodiversity. Real ecosystems often involve competitions among more than three species. By extending the RPS game model to five (rock-paper-scissors-lizard-Spock, or RPSLS) mobile species, we uncover a fundamental type of mesoscopic interactions among subgroups of species. In particular, competitions at the microscopic level lead to the emergence of various local groups in different regions of the space, each involving three species. It is the interactions among the groups that fundamentally determine how many species can coexist. In fact, as the mobility is increased from zero, two transitions can occur: one from a five- to a three-species coexistence state and another from the latter to a uniform, single-species state. We develop a mean-field theory to show that, in order to understand the first transition, group interactions at the mesoscopic scale must be taken into account. Our findings suggest, more broadly, the importance of mesoscopic interactions in coexistence of great many species.
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34
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Szolnoki A, Mobilia M, Jiang LL, Szczesny B, Rucklidge AM, Perc M. Cyclic dominance in evolutionary games: a review. J R Soc Interface 2014; 11:20140735. [PMID: 25232048 PMCID: PMC4191105 DOI: 10.1098/rsif.2014.0735] [Citation(s) in RCA: 143] [Impact Index Per Article: 14.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/07/2014] [Accepted: 08/22/2014] [Indexed: 11/12/2022] Open
Abstract
Rock is wrapped by paper, paper is cut by scissors and scissors are crushed by rock. This simple game is popular among children and adults to decide on trivial disputes that have no obvious winner, but cyclic dominance is also at the heart of predator-prey interactions, the mating strategy of side-blotched lizards, the overgrowth of marine sessile organisms and competition in microbial populations. Cyclical interactions also emerge spontaneously in evolutionary games entailing volunteering, reward, punishment, and in fact are common when the competing strategies are three or more, regardless of the particularities of the game. Here, we review recent advances on the rock-paper-scissors (RPS) and related evolutionary games, focusing, in particular, on pattern formation, the impact of mobility and the spontaneous emergence of cyclic dominance. We also review mean-field and zero-dimensional RPS models and the application of the complex Ginzburg-Landau equation, and we highlight the importance and usefulness of statistical physics for the successful study of large-scale ecological systems. Directions for future research, related, for example, to dynamical effects of coevolutionary rules and invasion reversals owing to multi-point interactions, are also outlined.
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Affiliation(s)
- Attila Szolnoki
- Institute of Technical Physics and Materials Science, Research Centre for Natural Sciences, Hungarian Academy of Sciences, PO Box 49, 1525 Budapest, Hungary
| | - Mauro Mobilia
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
| | - Luo-Luo Jiang
- College of Physics and Electronic Information Engineering, Wenzhou University, 325035 Wenzhou, People's Republic of China
| | - Bartosz Szczesny
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
| | - Alastair M Rucklidge
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
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35
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Rulands S, Jahn D, Frey E. Specialization and bet hedging in heterogeneous populations. PHYSICAL REVIEW LETTERS 2014; 113:108102. [PMID: 25238387 DOI: 10.1103/physrevlett.113.108102] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/04/2014] [Indexed: 06/03/2023]
Abstract
Phenotypic heterogeneity is a strategy commonly used by bacteria to rapidly adapt to changing environmental conditions. Here, we study the interplay between phenotypic heterogeneity and genetic diversity in spatially extended populations. By analyzing the spatiotemporal dynamics, we show that the level of mobility and the type of competition qualitatively influence the persistence of phenotypic heterogeneity. While direct competition generally promotes persistence of phenotypic heterogeneity, specialization dominates in models with indirect competition irrespective of the degree of mobility.
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Affiliation(s)
- Steffen Rulands
- Department of Physics, Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, Theresienstrasse 37, D-80333 Munich, Germany
| | - David Jahn
- Department of Physics, Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, Theresienstrasse 37, D-80333 Munich, Germany
| | - Erwin Frey
- Department of Physics, Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, Theresienstrasse 37, D-80333 Munich, Germany
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Szczesny B, Mobilia M, Rucklidge AM. Characterization of spiraling patterns in spatial rock-paper-scissors games. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:032704. [PMID: 25314470 DOI: 10.1103/physreve.90.032704] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2014] [Indexed: 06/04/2023]
Abstract
The spatiotemporal arrangement of interacting populations often influences the maintenance of species diversity and is a subject of intense research. Here, we study the spatiotemporal patterns arising from the cyclic competition between three species in two dimensions. Inspired by recent experiments, we consider a generic metapopulation model comprising "rock-paper-scissors" interactions via dominance removal and replacement, reproduction, mutations, pair exchange, and hopping of individuals. By combining analytical and numerical methods, we obtain the model's phase diagram near its Hopf bifurcation and quantitatively characterize the properties of the spiraling patterns arising in each phase. The phases characterizing the cyclic competition away from the Hopf bifurcation (at low mutation rate) are also investigated. Our analytical approach relies on the careful analysis of the properties of the complex Ginzburg-Landau equation derived through a controlled (perturbative) multiscale expansion around the model's Hopf bifurcation. Our results allow us to clarify when spatial "rock-paper-scissors" competition leads to stable spiral waves and under which circumstances they are influenced by nonlinear mobility.
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Affiliation(s)
- Bartosz Szczesny
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
| | - Mauro Mobilia
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
| | - Alastair M Rucklidge
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
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37
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Szolnoki A, Vukov J, Perc M. From pairwise to group interactions in games of cyclic dominance. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:062125. [PMID: 25019743 DOI: 10.1103/physreve.89.062125] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/10/2014] [Indexed: 06/03/2023]
Abstract
We study the rock-paper-scissors game in structured populations, where the invasion rates determine individual payoffs that govern the process of strategy change. The traditional version of the game is recovered if the payoffs for each potential invasion stem from a single pairwise interaction. However, the transformation of invasion rates to payoffs also allows the usage of larger interaction ranges. In addition to the traditional pairwise interaction, we therefore consider simultaneous interactions with all nearest neighbors, as well as with all nearest and next-nearest neighbors, thus effectively going from single pair to group interactions in games of cyclic dominance. We show that differences in the interaction range affect not only the stationary fractions of strategies but also their relations of dominance. The transition from pairwise to group interactions can thus decelerate and even revert the direction of the invasion between the competing strategies. Like in evolutionary social dilemmas, in games of cyclic dominance, too, the indirect multipoint interactions that are due to group interactions hence play a pivotal role. Our results indicate that, in addition to the invasion rates, the interaction range is at least as important for the maintenance of biodiversity among cyclically competing strategies.
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Affiliation(s)
- Attila Szolnoki
- Institute of Technical Physics and Materials Science, Research Centre for Natural Sciences, Hungarian Academy of Sciences, P. O. Box 49, H-1525 Budapest, Hungary
| | - Jeromos Vukov
- Institute of Technical Physics and Materials Science, Research Centre for Natural Sciences, Hungarian Academy of Sciences, P. O. Box 49, H-1525 Budapest, Hungary
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia
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Avelino PP, Bazeia D, Losano L, Menezes J, de Oliveira BF. Interfaces with internal structures in generalized rock-paper-scissors models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:042710. [PMID: 24827281 DOI: 10.1103/physreve.89.042710] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/06/2013] [Indexed: 06/03/2023]
Abstract
In this work we investigate the development of stable dynamical structures along interfaces separating domains belonging to enemy partnerships in the context of cyclic predator-prey models with an even number of species N≥8. We use both stochastic and field theory simulations in one and two spatial dimensions, as well as analytical arguments, to describe the association at the interfaces of mutually neutral individuals belonging to enemy partnerships and to probe their role in the development of the dynamical structures at the interfaces. We identify an interesting behavior associated with the symmetric or asymmetric evolution of the interface profiles depending on whether N/2 is odd or even, respectively. We also show that the macroscopic evolution of the interface network is not very sensitive to the internal structure of the interfaces. Although this work focuses on cyclic predator-prey models with an even number of species, we argue that the results are expected to be quite generic in the context of spatial stochastic May-Leonard models.
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Affiliation(s)
- P P Avelino
- Centro de Astrofísica da Universidade do Porto, 4150-762 Porto, Portugal and Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, 4169-007 Porto, Portugal
| | - D Bazeia
- Departamento de Física, Universidade Federal da Paraíba 58051-970 João Pessoa, PB, Brazil
| | - L Losano
- Departamento de Física, Universidade Federal da Paraíba 58051-970 João Pessoa, PB, Brazil
| | - J Menezes
- Escola de Ciências e Tecnologia, Universidade Federal do Rio Grande do Norte Caixa Postal 1524, 59072-970, Natal, RN, Brazil and Institute for Biodiversity and Ecosystem Dynamics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands
| | - B F de Oliveira
- Departamento de Física, Universidade Estadual de Maringá, 87020-900 Maringá, PR, Brazil
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Vukov J, Szolnoki A, Szabó G. Diverging fluctuations in a spatial five-species cyclic dominance game. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:022123. [PMID: 24032791 DOI: 10.1103/physreve.88.022123] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/06/2013] [Indexed: 06/02/2023]
Abstract
A five-species predator-prey model is studied on a square lattice where each species has two prey and two predators on the analogy to the rock-paper-scissors-lizard-Spock game. The evolution of the spatial distribution of species is governed by site exchange and invasion between the neighboring predator-prey pairs, where the cyclic symmetry can be characterized by two different invasion rates. The mean-field analysis has indicated periodic oscillations in the species densities with a frequency becoming zero for a specific ratio of invasion rates. When varying the ratio of invasion rates, the appearance of this zero-eigenvalue mode is accompanied by neutrality between the species associations. Monte Carlo simulations of the spatial system reveal diverging fluctuations at a specific invasion rate, which can be related to the vanishing dominance between all pairs of species associations.
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Affiliation(s)
- Jeromos Vukov
- Institute of Technical Physics and Materials Science, Research Centre for Natural Sciences, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
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40
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Guisoni NC, Loscar ES, Girardi M. Phase diagram of a cyclic predator-prey model with neutral-pair exchange. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:022133. [PMID: 24032801 DOI: 10.1103/physreve.88.022133] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/16/2013] [Indexed: 06/02/2023]
Abstract
In this paper we obtain the phase diagram of a four-species predator-prey lattice model by using the proposed gradient method. We consider cyclic transitions between consecutive states, representing invasion or predation, and allowed the exchange between neighboring neutral pairs. By applying a gradient in the invasion rate parameter one can see, in the same simulation, the presence of two symmetric absorbing phases, composed by neutral pairs, and an active phase that includes all four species. In this sense, the study of a single-valued interface and its fluctuations give the critical point of the irreversible phase transition and the corresponding universality classes. Also, the consideration of a multivalued interface and its fluctuations bring the percolation threshold. We show that the model presents two lines of irreversible first-order phase transition between the two absorbing phases and the active phase. Depending on the value of the system parameters, these lines can converge into a triple point, which is the beginning of a first-order irreversible line between the two absorbing phases, or end in two critical points belonging to the directed percolation universality class. Standard simulations for some characteristic values of the parameters confirm the order of the transitions as determined by the gradient method. Besides, below the triple point the model presents two standard percolation lines in the active phase and above a first-order percolation transition as already found in other similar models.
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Affiliation(s)
- Nara C Guisoni
- Instituto de Física de Líquidos y Sistemas Biológicos (IFLYSIB), Universidad Nacional de La Plata, CONICET CCT-La Plata, Calle 59-789 (1900) La Plata, Argentina
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41
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Park J, Do Y, Huang ZG, Lai YC. Persistent coexistence of cyclically competing species in spatially extended ecosystems. CHAOS (WOODBURY, N.Y.) 2013; 23:023128. [PMID: 23822493 DOI: 10.1063/1.4811298] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
A fundamental result in the evolutionary-game paradigm of cyclic competition in spatially extended ecological systems, as represented by the classic Reichenbach-Mobilia-Frey (RMF) model, is that high mobility tends to hamper or even exclude species coexistence. This result was obtained under the hypothesis that individuals move randomly without taking into account the suitability of their local environment. We incorporate local habitat suitability into the RMF model and investigate its effect on coexistence. In particular, we hypothesize the use of "basic instinct" of an individual to determine its movement at any time step. That is, an individual is more likely to move when the local habitat becomes hostile and is no longer favorable for survival and growth. We show that, when such local habitat suitability is taken into account, robust coexistence can emerge even in the high-mobility regime where extinction is certain in the RMF model. A surprising finding is that coexistence is accompanied by the occurrence of substantial empty space in the system. Reexamination of the RMF model confirms the necessity and the important role of empty space in coexistence. Our study implies that adaptation/movements according to local habitat suitability are a fundamental factor to promote species coexistence and, consequently, biodiversity.
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Affiliation(s)
- Junpyo Park
- Department of Mathematics, Kyungpook National University, Daegu 702-701, South Korea
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42
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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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43
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Greulich P, Waclaw B, Allen RJ. Mutational pathway determines whether drug gradients accelerate evolution of drug-resistant cells. PHYSICAL REVIEW LETTERS 2012; 109:088101. [PMID: 23002776 DOI: 10.1103/physrevlett.109.088101] [Citation(s) in RCA: 62] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/02/2012] [Indexed: 05/28/2023]
Abstract
Drug gradients are believed to play an important role in the evolution of bacteria resistant to antibiotics and tumors resistant to anticancer drugs. We use a statistical physics model to study the evolution of a population of malignant cells exposed to drug gradients, where drug resistance emerges via a mutational pathway involving multiple mutations. We show that a nonuniform drug distribution has the potential to accelerate the emergence of resistance when the mutational pathway involves a long sequence of mutants with increasing resistance, but if the pathway is short or crosses a fitness valley, the evolution of resistance may actually be slowed down by drug gradients. These predictions can be verified experimentally, and may help to improve strategies for combating the emergence of resistance.
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Affiliation(s)
- Philip Greulich
- SUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom
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44
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Lamouroux D, Eule S, Geisel T, Nagler J. Discriminating the effects of spatial extent and population size in cyclic competition among species. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:021911. [PMID: 23005789 DOI: 10.1103/physreve.86.021911] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/10/2011] [Revised: 07/25/2012] [Indexed: 06/01/2023]
Abstract
We introduce a population model for species under cyclic competition. This model allows individuals to coexist and interact on single cells while migration takes place between adjacent cells. In contrast to the model introduced by Reichenbach, Mobilia, and Frey [Reichenbach, Mobilia, and Frey, Nature (London) 448, 1046 (2007)], we find that the emergence of spirals results in an ambiguous behavior regarding the stability of coexistence. The typical time until extinction exhibits, however, a qualitatively opposite dependence on the newly introduced nonunit carrying capacity in the spiraling and the nonspiraling regimes. This allows us to determine a critical mobility that marks the onset of this spiraling state sharply. In contrast, we demonstrate that the conventional finite size stability analysis with respect to spatial size is of limited use for identifying the onset of the spiraling regime.
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Affiliation(s)
- D Lamouroux
- Max Planck Institute for Dynamics & Self-Organization, 37077 Göttingen, Germany
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45
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Makeev AG, Kurkina ES, Kevrekidis IG. Kinetic Monte Carlo simulations of travelling pulses and spiral waves in the lattice Lotka-Volterra model. CHAOS (WOODBURY, N.Y.) 2012; 22:023141. [PMID: 22757548 DOI: 10.1063/1.4729141] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
Kinetic Monte Carlo simulations are used to study the stochastic two-species Lotka-Volterra model on a square lattice. For certain values of the model parameters, the system constitutes an excitable medium: travelling pulses and rotating spiral waves can be excited. Stable solitary pulses travel with constant (modulo stochastic fluctuations) shape and speed along a periodic lattice. The spiral waves observed persist sometimes for hundreds of rotations, but they are ultimately unstable and break-up (because of fluctuations and interactions between neighboring fronts) giving rise to complex dynamic behavior in which numerous small spiral waves rotate and interact with each other. It is interesting that travelling pulses and spiral waves can be exhibited by the model even for completely immobile species, due to the non-local reaction kinetics.
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Affiliation(s)
- Alexei G Makeev
- Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow 119991, Russia.
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46
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Han TA, Traulsen A, Gokhale CS. On equilibrium properties of evolutionary multi-player games with random payoff matrices. Theor Popul Biol 2012; 81:264-72. [PMID: 22406614 DOI: 10.1016/j.tpb.2012.02.004] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/12/2011] [Revised: 02/07/2012] [Accepted: 02/16/2012] [Indexed: 10/28/2022]
Abstract
The analysis of equilibrium points in biological dynamical systems has been of great interest in a variety of mathematical approaches to biology, such as population genetics, theoretical ecology or evolutionary game theory. The maximal number of equilibria and their classification based on stability have been the primary subjects of these studies, for example in the context of two-player games with multiple strategies. Herein, we address a different question using evolutionary game theory as a tool. If the payoff matrices are drawn randomly from an arbitrary distribution, what are the probabilities of observing a certain number of (stable) equilibria? We extend the domain of previous results for the two-player framework, which corresponds to a single diploid locus in population genetics, by addressing the full complexity of multi-player games with multiple strategies. In closing, we discuss an application and illustrate how previous results on the number of equilibria, such as the famous Feldman-Karlin conjecture on the maximal number of isolated fixed points in a viability selection model, can be obtained as special cases of our results based on multi-player evolutionary games. We also show how the probability of realizing a certain number of equilibria changes as we increase the number of players and number of strategies.
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Affiliation(s)
- The Anh Han
- Center of Artificial Intelligence, Department of Informatics, Faculty of Science and Technologies, New University of Lisbon, P-2829-516 Caparica, Portugal
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47
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Jiang LL, Zhou T, Perc M, Wang BH. Effects of competition on pattern formation in the rock-paper-scissors game. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:021912. [PMID: 21929025 DOI: 10.1103/physreve.84.021912] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/15/2011] [Indexed: 05/31/2023]
Abstract
We investigate the impact of cyclic competition on pattern formation in the rock-paper-scissors game. By separately considering random and prepared initial conditions, we observe a critical influence of the competition rate p on the stability of spiral waves and on the emergence of biodiversity. In particular, while increasing values of p promote biodiversity, they may act detrimentally on spatial pattern formation. For random initial conditions, we observe a phase transition from biodiversity to an absorbing phase, whereby the critical value of mobility grows linearly with increasing values of p on a log-log scale but then saturates as p becomes large. For prepared initial conditions, we observe the formation of single-armed spirals, but only for values of p that are below a critical value. Once above that value, the spirals break up and form disordered spatial structures, mainly because of the percolation of vacant sites. Thus there exists a critical value of the competition rates p(c) for stable single-armed spirals in finite populations. Importantly though, p(c) increases with increasing system size because noise reinforces the disintegration of ordered patterns. In addition, we also find that p(c) increases with the mobility. These phenomena are reproduced by a deterministic model that is based on nonlinear partial differential equations. Our findings indicate that competition is vital for the sustenance of biodiversity and the emergence of pattern formation in ecosystems governed by cyclical interactions.
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Affiliation(s)
- Luo-Luo Jiang
- College of Physics and Electronic Information Engineering, Wenzhou University, Wenzhou 325027, China
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48
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Durney CH, Case SO, Pleimling M, Zia RKP. Saddles, arrows, and spirals: deterministic trajectories in cyclic competition of four species. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:051108. [PMID: 21728491 DOI: 10.1103/physreve.83.051108] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/23/2011] [Indexed: 05/31/2023]
Abstract
Population dynamics in systems composed of cyclically competing species has been of increasing interest recently. Here we investigate a system with four or more species. Using mean field theory, we study in detail the trajectories in configuration space of the population fractions. We discover a variety of orbits, shaped like saddles, spirals, and straight lines. Many of their properties are found explicitly. Most remarkably, we identify a collective variable that evolves simply as an exponential: Q ∝ e(λt), where λ is a function of the reaction rates. It provides information on the state of the system for late times (as well as for t→-∞). We discuss implications of these results for the evolution of a finite, stochastic system. A generalization to an arbitrary number of cyclically competing species yields valuable insights into universal properties of such systems.
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Affiliation(s)
- C H Durney
- Department of Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA
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49
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Wang WX, Ni X, Lai YC, Grebogi C. Pattern formation, synchronization, and outbreak of biodiversity in cyclically competing games. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:011917. [PMID: 21405723 PMCID: PMC3654227 DOI: 10.1103/physreve.83.011917] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2010] [Revised: 11/16/2010] [Indexed: 05/30/2023]
Abstract
Species in nature are typically mobile over diverse distance scales, examples of which range from bacteria run to long-distance animal migrations. These behaviors can have a significant impact on biodiversity. Addressing the role of migration in biodiversity microscopically is fundamental but remains a challenging problem in interdisciplinary science. We incorporate both intra- and inter-patch migrations in stochastic games of cyclic competitions and find that the interplay between the migrations at the local and global scales can lead to robust species coexistence characterized dynamically by the occurrence of remarkable target-wave patterns in the absence of any external control. The waves can emerge from either mixed populations or isolated species in different patches, regardless of the size and the location of the migration target. We also find that, even in a single-species system, target waves can arise from rare mutations, leading to an outbreak of biodiversity. A surprising phenomenon is that target waves in different patches can exhibit synchronization and time-delayed synchronization, where the latter potentially enables the prediction of future evolutionary dynamics. We provide a physical theory based on the spatiotemporal organization of the target waves to explain the synchronization phenomena. We also investigate the basins of coexistence and extinction to establish the robustness of biodiversity through migrations. Our results are relevant to issues of general and broader interest such as pattern formation, control in excitable systems, and the origin of order arising from self-organization in social and natural systems.
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Affiliation(s)
- Wen-Xu Wang
- School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
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50
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Ni X, Yang R, Wang WX, Lai YC, Grebogi C. Basins of coexistence and extinction in spatially extended ecosystems of cyclically competing species. CHAOS (WOODBURY, N.Y.) 2010; 20:045116. [PMID: 21198128 DOI: 10.1063/1.3526993] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
Microscopic models based on evolutionary games on spatially extended scales have recently been developed to address the fundamental issue of species coexistence. In this pursuit almost all existing works focus on the relevant dynamical behaviors originated from a single but physically reasonable initial condition. To gain comprehensive and global insights into the dynamics of coexistence, here we explore the basins of coexistence and extinction and investigate how they evolve as a basic parameter of the system is varied. Our model is cyclic competitions among three species as described by the classical rock-paper-scissors game, and we consider both discrete lattice and continuous space, incorporating species mobility and intraspecific competitions. Our results reveal that, for all cases considered, a basin of coexistence always emerges and persists in a substantial part of the parameter space, indicating that coexistence is a robust phenomenon. Factors such as intraspecific competition can, in fact, promote coexistence by facilitating the emergence of the coexistence basin. In addition, we find that the extinction basins can exhibit quite complex structures in terms of the convergence time toward the final state for different initial conditions. We have also developed models based on partial differential equations, which yield basin structures that are in good agreement with those from microscopic stochastic simulations. To understand the origin and emergence of the observed complicated basin structures is challenging at the present due to the extremely high dimensional nature of the underlying dynamical system.
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Affiliation(s)
- Xuan Ni
- School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
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