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Diaz-Ruelas A, Jeldtoft Jensen H, Piovani D, Robledo A. Tangent map intermittency as an approximate analysis of intermittency in a high dimensional fully stochastic dynamical system: The Tangled Nature model. CHAOS (WOODBURY, N.Y.) 2016; 26:123105. [PMID: 28039973 DOI: 10.1063/1.4968207] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
It is well known that low-dimensional nonlinear deterministic maps close to a tangent bifurcation exhibit intermittency and this circumstance has been exploited, e.g., by Procaccia and Schuster [Phys. Rev. A 28, 1210 (1983)], to develop a general theory of 1/f spectra. This suggests it is interesting to study the extent to which the behavior of a high-dimensional stochastic system can be described by such tangent maps. The Tangled Nature (TaNa) Model of evolutionary ecology is an ideal candidate for such a study, a significant model as it is capable of reproducing a broad range of the phenomenology of macroevolution and ecosystems. The TaNa model exhibits strong intermittency reminiscent of punctuated equilibrium and, like the fossil record of mass extinction, the intermittency in the model is found to be non-stationary, a feature typical of many complex systems. We derive a mean-field version for the evolution of the likelihood function controlling the reproduction of species and find a local map close to tangency. This mean-field map, by our own local approximation, is able to describe qualitatively only one episode of the intermittent dynamics of the full TaNa model. To complement this result, we construct a complete nonlinear dynamical system model consisting of successive tangent bifurcations that generates time evolution patterns resembling those of the full TaNa model in macroscopic scales. The switch from one tangent bifurcation to the next in the sequences produced in this model is stochastic in nature, based on criteria obtained from the local mean-field approximation, and capable of imitating the changing set of types of species and total population in the TaNa model. The model combines full deterministic dynamics with instantaneous parameter random jumps at stochastically drawn times. In spite of the limitations of our approach, which entails a drastic collapse of degrees of freedom, the description of a high-dimensional model system in terms of a low-dimensional one appears to be illuminating.
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Affiliation(s)
- Alvaro Diaz-Ruelas
- Instituto de Física, Universidad Nacional Autónoma de México, Ciudad Universitaria, Ciudad de México 04510, Mexico
| | - Henrik Jeldtoft Jensen
- Centre for Complexity Science and Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom
| | - Duccio Piovani
- Centre for Complexity Science and Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom
| | - Alberto Robledo
- Instituto de Física y Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México, Ciudad Universitaria, Ciudad de México 04510, Mexico
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Murase Y, Shimada T, Ito N, Rikvold PA. Random walk in genome space: a key ingredient of intermittent dynamics of community assembly on evolutionary time scales. J Theor Biol 2010; 264:663-72. [PMID: 20362586 DOI: 10.1016/j.jtbi.2010.03.043] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/19/2009] [Revised: 03/29/2010] [Accepted: 03/29/2010] [Indexed: 11/17/2022]
Abstract
Community assembly is studied using individual-based multispecies models. The models have stochastic population dynamics with mutation, migration, and extinction of species. Mutants appear as a result of mutation of the resident species, while migrants have no correlation with the resident species. It is found that the dynamics of community assembly with mutations are quite different from the case with migrations. In contrast to mutation models, which show intermittent dynamics of quasi-steady states interrupted by sudden reorganizations of the community, migration models show smooth and gradual renewal of the community. As a consequence, instead of the 1/f diversity fluctuations found for the mutation models, 1/f(2), random-walk like fluctuations are observed for the migration models. In addition, a characteristic species-lifetime distribution is found: a power law that is cut off by a "skewed" distribution in the long-lifetime regime. The latter has a longer tail than a simple exponential function, which indicates an age-dependent species-mortality function. Since this characteristic profile has been observed, both in fossil data and in several other mathematical models, we conclude that it is a universal feature of macroevolution.
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Affiliation(s)
- Yohsuke Murase
- Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan.
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Murase Y, Shimada T, Ito N, Rikvold PA. Effects of demographic stochasticity on biological community assembly on evolutionary time scales. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:041908. [PMID: 20481754 DOI: 10.1103/physreve.81.041908] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/10/2009] [Revised: 01/25/2010] [Indexed: 05/29/2023]
Abstract
We study the effects of demographic stochasticity on the long-term dynamics of biological coevolution models of community assembly. The noise is induced in order to check the validity of deterministic population dynamics. While mutualistic communities show little dependence on the stochastic population fluctuations, predator-prey models show strong dependence on the stochasticity, indicating the relevance of the finiteness of the populations. For a predator-prey model, the noise causes drastic decreases in diversity and total population size. The communities that emerge under influence of the noise consist of species strongly coupled with each other and have stronger linear stability around the fixed-point populations than the corresponding noiseless model. The dynamics on evolutionary time scales for the predator-prey model are also altered by the noise. Approximate 1/f fluctuations are observed with noise, while 1/f2 fluctuations are found for the model without demographic noise.
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Affiliation(s)
- Yohsuke Murase
- Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
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Rikvold PA. Complex dynamics in coevolution models with ratio-dependent functional response. ECOLOGICAL COMPLEXITY 2009. [DOI: 10.1016/j.ecocom.2009.08.007] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Filotas E, Grant M, Parrott L, Rikvold PA. Community-driven dispersal in an individual-based predator–prey model. ECOLOGICAL COMPLEXITY 2008. [DOI: 10.1016/j.ecocom.2008.01.002] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Rikvold PA. Self-optimization, community stability, and fluctuations in two individual-based models of biological coevolution. J Math Biol 2007; 55:653-77. [PMID: 17534620 DOI: 10.1007/s00285-007-0101-y] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/25/2006] [Revised: 01/18/2007] [Indexed: 10/23/2022]
Abstract
We compare and contrast the long-time dynamical properties of two individual-based models of biological coevolution. Selection occurs via multispecies, stochastic population dynamics with reproduction probabilities that depend nonlinearly on the population densities of all species resident in the community. New species are introduced through mutation. Both models are amenable to exact linear stability analysis, and we compare the analytic results with large-scale kinetic Monte Carlo simulations, obtaining the population size as a function of an average interspecies interaction strength. Over time, the models self-optimize through mutation and selection to approximately maximize a community potential function, subject only to constraints internal to the particular model. If the interspecies interactions are randomly distributed on an interval including positive values, the system evolves toward self-sustaining, mutualistic communities. In contrast, for the predator-prey case the matrix of interactions is antisymmetric, and a nonzero population size must be sustained by an external resource. Time series of the diversity and population size for both models show approximate 1/f noise and power-law distributions for the lifetimes of communities and species. For the mutualistic model, these two lifetime distributions have the same exponent, while their exponents are different for the predator-prey model. The difference is probably due to greater resilience toward mass extinctions in the food-web like communities produced by the predator-prey model.
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Affiliation(s)
- Per Arne Rikvold
- School of Computational Science, Center for Materials Research and Technology, National High Magnetic Field Laboratory, and Department of Physics, Florida State University, Tallahassee, FL 32306-4120, USA.
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Rikvold PA, Sevim V. Individual-based predator-prey model for biological coevolution: fluctuations, stability, and community structure. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:051920. [PMID: 17677111 DOI: 10.1103/physreve.75.051920] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2006] [Revised: 04/12/2007] [Indexed: 05/16/2023]
Abstract
We study an individual-based predator-prey model of biological coevolution, using linear stability analysis and large-scale kinetic Monte Carlo simulations. The model exhibits approximate 1/f noise in diversity and population-size fluctuations, and it generates a sequence of quasisteady communities in the form of simple food webs. These communities are quite resilient toward the loss of one or a few species, which is reflected in different power-law exponents for the durations of communities and the lifetimes of species. The exponent for the former is near -1 , while the latter is close to -2 . Statistical characteristics of the evolving communities, including degree (predator and prey) distributions and proportions of basal, intermediate, and top species, compare reasonably with data for real food webs.
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Affiliation(s)
- Per Arne Rikvold
- School of Computational Science, Center for Materials Research and Technology, and Department of Physics, Florida State University, Tallahassee, FL 32306-4120, USA.
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The Tangled Nature model with inheritance and constraint: Evolutionary ecology restricted by a conserved resource. ECOLOGICAL COMPLEXITY 2006. [DOI: 10.1016/j.ecocom.2006.06.001] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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Lawson D, Jensen HJ. The species–area relationship and evolution. J Theor Biol 2006; 241:590-600. [PMID: 16458929 DOI: 10.1016/j.jtbi.2005.12.018] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/30/2005] [Revised: 12/21/2005] [Accepted: 12/27/2005] [Indexed: 10/25/2022]
Abstract
Models relating to the species-area curve usually assume the existence of species, and are concerned mainly with ecological timescales. We examine an individual-based model of co-evolution on a spatial lattice based on the tangled nature model in which species are emergent structures, and show that reproduction, mutation and dispersion by diffusion, with interaction via genotype space, produces power-law species-area relations as observed in ecological measurements at medium scales. We find that long-lasting co-evolutionary habitats form, allowing high diversity levels in a spatially homogenous system.
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Affiliation(s)
- Daniel Lawson
- Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK
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Sevim V, Rikvold PA. Effects of correlated interactions in a biological coevolution model with individual-based dynamics. ACTA ACUST UNITED AC 2005. [DOI: 10.1088/0305-4470/38/43/005] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Anderson PE, Jensen HJ. Network properties, species abundance and evolution in a model of evolutionary ecology. J Theor Biol 2005; 232:551-8. [PMID: 15588635 DOI: 10.1016/j.jtbi.2004.03.029] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2003] [Revised: 02/20/2004] [Accepted: 03/18/2004] [Indexed: 11/22/2022]
Abstract
We study the evolution of the network properties of a populated network embedded in a genotype space characterized by either a low or a high number of potential links, with particular emphasis on the connectivity and clustering. Evolution produces two distinct types of network. When a specific genotype is only able to influence a few other genotypes, the ecosystem consists of separate non-interacting clusters (i.e. isolated compartments) in genotype space. When different types may influence a large number of other sites, the network becomes one large interconnected cluster. The distribution of interaction strengths--but not the number of connections--changes significantly with time. We find that the species abundance is only realistic for a high level of species connectivity. This suggests that real ecosystems form one interconnected whole in which selection leads to stronger interactions between the different types. Analogies with niche and neutral theory and assembly models are also considered.
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Affiliation(s)
- Paul E Anderson
- Department of Mathematics, Imperial College, 180 Queen's Gate, London, SW7 2AZ, UK
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McKane AJ, Newman TJ. Stochastic models in population biology and their deterministic analogs. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:041902. [PMID: 15600430 DOI: 10.1103/physreve.70.041902] [Citation(s) in RCA: 99] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/18/2004] [Indexed: 05/08/2023]
Abstract
We introduce a class of stochastic population models based on "patch dynamics." The size of the patch may be varied, and this allows one to quantify the departures of these stochastic models from various mean-field theories, which are generally valid as the patch size becomes very large. These models may be used to formulate a broad range of biological processes in both spatial and nonspatial contexts. Here, we concentrate on two-species competition. We present both a mathematical analysis of the patch model, in which we derive the precise form of the competition mean-field equations (and their first-order corrections in the nonspatial case), and simulation results. These mean-field equations differ, in some important ways, from those which are normally written down on phenomenological grounds. Our general conclusion is that mean-field theory is more robust for spatial models than for a single isolated patch. This is due to the dilution of stochastic effects in a spatial setting resulting from repeated rescue events mediated by interpatch diffusion. However, discrete effects due to modest patch sizes lead to striking deviations from mean-field theory even in a spatial setting.
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Affiliation(s)
- A J McKane
- Department of Theoretical Physics, University of Manchester, Manchester M13 9PL, United Kingdom
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