DOCSLIB.ORG

Collective Dynamics and Homeostatic Emergence in Complex Adaptive Ecosystem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Collective Dynamics and Homeostatic Emergence in Complex Adaptive Ecosystem ECAL - General Track Collective Dynamics and Homeostatic Emergence in Complex Adaptive Ecosystem Dharani Punithan and RI (Bob) McKay Structural Complexity Laboratory, College of Engineering, School of Computer Science and Engineering, Seoul National University, South Korea punithan.dharani,rimsnucse @gmail.com { } Downloaded from http://direct.mit.edu/isal/proceedings-pdf/ecal2013/25/332/1901807/978-0-262-31709-2-ch049.pdf by guest on 26 September 2021 Abstract 1. Life-environment feedback via the daisyworld model We investigate the behaviour of the daisyworld model on 2. State-topology feedback via an adaptive network model an adaptive network, comparing it to previous studies on a fixed topology grid, and a fixed small-world (Newman-Watts 3. Density-growth feedback via a logistic growth model (NW)) network. The adaptive networks eventually generate topologies with small-world effect behaving similarly to the The topology of the our ecosystem evolves with a simple NW model – and radically different from the grid world. Un- der the same parameter settings, static but complex patterns local rule – a frozen habitat is reciprocally linked to an ac- emerge in the grid world. In the NW model, we see the tive habitat – and self-organises to complex topologies with emergence of completely coherent periodic dominance. In small-world effect. In this paper, we focus on the emergent the adaptive-topology world, the systems may transit through collective phenomena and properties that arise in egalitarian varied behaviours, but can self-organise to a small-world small-world ecosystems, constructed from a large number of network structure with similar cyclic behaviour to the NW model. interacting adaptively linked habitats. Background Introduction Our model has three feedback loops determining its dynam- In this paper, we examine connectivity changes in a com- ics. We next detail the relevant background. plex adaptive ecosystem based on the daisyworld model, combining coupled map lattice (CML) and complex adap- Daisyworld (homeostatic self-regulation of the environ- tive network models. Daisyworld, proposed by Watson and ment by the biota) Daisyworld (Watson and Lovelock, Lovelock (1983), is a simple mathematical system demon- 1983) is an imaginary planet where only two types of species strating planetary homeostasis – self-regulation of the envi- live – black and white daisies. These biotic components in- ronment by biota and self-sustainability of life through inter- teract stigmergically via an abiotic component – tempera- action with the environment. Daisyworld topologies in the ture. The different colours of the daisies influence the albedo literature are static, with only local connections (Wood et al., (reflectivity) of the planet. In the beginning, the atmosphere 2008). In our previous work (Punithan et al., 2011; Punithan of the daisyworld is cooler and only black daisies thrive as and McKay, 2013), we have investigated ecological home- they absorb all the energy. As the black daisy population ostasis in preconstructed static topologies with local and expands, it warms the planet. When it is too warm for black non-local long range couplings –small-world networks. But daisies to survive, white daisies start to bloom since they complex networks in nature and society are adaptive, in that reflect all the energy back into space. As the white daisy they exhibit feedback between the local dynamics of nodes cover spreads, it cools the planet. When it is too cold for (state) and the evolution of the topological structure (Gross the survival of white daisies, again black daisies thrive. This and Blasius, 2008; Gross and Sayama, 2009). Examples in- endless cycle, owing to the bi-directional feedback loop be- clude genetic, neural, immunity, ecological, economic and tween life and the environment, self-regulates the tempera- social networks, complex game interactions etc. ture and thereby allows life to persist. The topology of our ecosystem evolves in response to lo- cal habitat states, and the evolved topology in turn impacts Adaptive Networks (dynamics on the network interact- the habitat states. Our adaptive and self-maintaining ecosys- ing with dynamics of the network) In most real-world tem, based on CML consists of a set of diffusively cou- networks, the topology itself is a dynamical system which pled habitats incorporating logistic growth of life with bi- changes in time and in response to the dynamics of the states directional biota-environment influences. Thus our ecosys- of the nodes (dynamics of the network). The evolved topol- tem incorporates three kinds of feedback: ogy in turn influences the dynamics of the states of the nodes ECAL 2013 332 ECAL - General Track (dynamics on the network), creating a feedback loop be- Denoting the set of neighbours of (i, j) as < l, m >, we tween the dynamics of the nodes and the evolution of the can simplify equation 2 to: topology. Networks exhibiting such a feedback loop (mu- tual evolution of structure and state values) are called adap- ǫ k x = f (1 ǫ)x + x (3) tive or coevolutionary networks (Gross and Blasius, 2008; (i,j,t+1) − (i,j,t) k (l,m,t) Gross and Sayama, 2009). In road networks, the topology <l,m> of the road influences the traffic flow, while traffic conges- Small-world Phenomena The co-occurence of high clus- tion influences the construction of new roads. In the vascu- tering (as in regular networks) and low characteristic path lar system, the topology of the blood vessels controls blood length (as in random networks) define a small-world struc- flow, while restrictions in blood flow influence the formation ture (Watts and Strogatz, 1998). These small-world network of new arteries (arteriogenesis). Numerous other examples properties, giving rise to the well-known “six degrees of are discussed in Gross and Blasius (2008). separation” phenomenon (Milgram, 1967), are quantified by Downloaded from http://direct.mit.edu/isal/proceedings-pdf/ecal2013/25/332/1901807/978-0-262-31709-2-ch049.pdf by guest on 26 September 2021 Logistic Growth Model (density-dependent growth rate) two statistical measures: the clustering co-efficient (measur- The discretised logistic growth model (Verhulst model) is ing local cliqui-ness) , and the characteristic path length C L¯ key to population ecology. (measuring global connectedness). Their average values and ¯ for a network with n nodes are defined by: C L Pt P = P 1+r 1 ; r>0,κ>0 (1) n (t+1) t − κ 1 E(Γ ) ¯ = | v | (4) C n k where P [0,κ] is the population size (at times t and t+1), v=1 v ∈ 2 r is the intrinsic growth rate (bifurcation parameter), κ is the carrying capacity (maximum sustainable population be- where Γ is the neighbourhood of a node v, E(Γ ) is the v | v | yond which P cannot increase). The parameter r amplifies number of actual links in the neighbourhood of v, kv is the P population growth and the component [1 t ] dampens the number of nodes in the subnetwork Γ and kv is the num- − κ v 2 growth due to over crowding. Thus population density self- ber of possible links in Γv; and regulates population growth rate. It is also well-known that n n chaos emerges from this growth model (May, 1976) in spite 1 ¯ = d (5) of the built-in regulatory mechanism. L n uv u=1 v>u 2 Coupled Map Lattice The coupled map lattice (Kaneko, 1985, 1992; Kaneko and Tsuda, 2001) incorporates discrete where duv is the shortest path between a pair of nodes u, v. time evolution (map) in a discrete space (lattice or network) as in cellular automata (CA), but takes continuous state val- Degree Distribution The degree of a node is the number ues as in partial differential equation (PDE) models. CML is of neighbours it is connected to. The degree distribution governed by the temporal nonlinear reaction (maps - f) and is defined as the normalised frequency distribution of de- the spatial diffusion (coupling - ǫ). grees over the whole network. The degree distribution of If f(x) is a reaction function of a dynamical variable (x), a network is a simple property which helps to classify net- the update of the variable is computed by combining that works. The regular network with Moore neighbourhoods reaction with discrete Laplacian diffusion. For a regular net- have the same degree (k =8) for all the nodes. The degree work with Moore neighbourhoods (k = 8), the update of x distribution of small-world networks (p in the small-world is computed as: regime (Punithan and McKay, 2013) follows a Poisson dis- tribution with exponential tail. Networks in which most ǫ nodes have approximately the same number of neighbours x = f (1 ǫ)x + x (i,j,t+1) − (i,j,t) 8 (i+1,j,t) are known as “egalitarian” networks (Buchanan, 2003). +x(i 1,j,t) + x(i,j+1,t) + x(i,j 1,t) − − (2) Model +x(i 1,j 1,t) + x(i+1,j 1,t) − − − Our ecosystem is a complex dynamic system in which +x(i 1,j+1,t) + x(i+1,j+1,t) the continuous state habitats diffusively interact with their − neighbours (coupled), evolve in discrete time (map) and are where x(i,j,t) is the spatio-temporal distribution of a dynam- distributed on a discrete space (lattice). Initially, we con- ical variable, ǫ [0, 1] is the coupling parameter (diffusion struct a 2-lattice with Moore neighbourhoods and periodic ∈ ′ rate), k is the number of interacting neighbours, f(x(i,j,t)) boundary conditions. Each point in the lattice represents a ′ is a local non-linear function and x(i,j,t) is the value after habitat with a maximum carrying capacity of 10, 000 daisies. diffusion. Each habitat in our ecosystem is a system.
Load more

Recommended publications
  • A General Representation of Dynamical Systems for Reservoir Computing
    A general representation of dynamical systems for reservoir computing Sidney Pontes-Filho∗,y,x, Anis Yazidi∗, Jianhua Zhang∗, Hugo Hammer∗, Gustavo B. M. Mello∗, Ioanna Sandvigz, Gunnar Tuftey and Stefano Nichele∗ ∗Department of Computer Science, Oslo Metropolitan University, Oslo, Norway yDepartment of Computer Science, Norwegian University of Science and Technology, Trondheim, Norway zDepartment of Neuromedicine and Movement Science, Norwegian University of Science and Technology, Trondheim, Norway Email: [email protected] Abstract—Dynamical systems are capable of performing com- TABLE I putation in a reservoir computing paradigm. This paper presents EXAMPLES OF DYNAMICAL SYSTEMS. a general representation of these systems as an artificial neural network (ANN). Initially, we implement the simplest dynamical Dynamical system State Time Connectivity system, a cellular automaton. The mathematical fundamentals be- Cellular automata Discrete Discrete Regular hind an ANN are maintained, but the weights of the connections Coupled map lattice Continuous Discrete Regular and the activation function are adjusted to work as an update Random Boolean network Discrete Discrete Random rule in the context of cellular automata. The advantages of such Echo state network Continuous Discrete Random implementation are its usage on specialized and optimized deep Liquid state machine Discrete Continuous Random learning libraries, the capabilities to generalize it to other types of networks and the possibility to evolve cellular automata and other dynamical systems in terms of connectivity, update and learning rules. Our implementation of cellular automata constitutes an reservoirs [6], [7]. Other systems can also exhibit the same initial step towards a general framework for dynamical systems. dynamics. The coupled map lattice [8] is very similar to It aims to evolve such systems to optimize their usage in reservoir CA, the only exception is that the coupled map lattice has computing and to model physical computing substrates.
    [Show full text]
  • Thermodynamic Properties of Coupled Map Lattices 1 Introduction
    Thermodynamic properties of coupled map lattices J´erˆome Losson and Michael C. Mackey Abstract This chapter presents an overview of the literature which deals with appli- cations of models framed as coupled map lattices (CML’s), and some recent results on the spectral properties of the transfer operators induced by various deterministic and stochastic CML’s. These operators (one of which is the well- known Perron-Frobenius operator) govern the temporal evolution of ensemble statistics. As such, they lie at the heart of any thermodynamic description of CML’s, and they provide some interesting insight into the origins of nontrivial collective behavior in these models. 1 Introduction This chapter describes the statistical properties of networks of chaotic, interacting el- ements, whose evolution in time is discrete. Such systems can be profitably modeled by networks of coupled iterative maps, usually referred to as coupled map lattices (CML’s for short). The description of CML’s has been the subject of intense scrutiny in the past decade, and most (though by no means all) investigations have been pri- marily numerical rather than analytical. Investigators have often been concerned with the statistical properties of CML’s, because a deterministic description of the motion of all the individual elements of the lattice is either out of reach or uninteresting, un- less the behavior can somehow be described with a few degrees of freedom. However there is still no consistent framework, analogous to equilibrium statistical mechanics, within which one can describe the probabilistic properties of CML’s possessing a large but finite number of elements.
    [Show full text]
  • Climate Modelling Primer
    A Climate Modelling Primer A Climate Modelling Primer, Third Edition. K. McGuffie and A. Henderson-Sellers. © 2005 John Wiley & Sons, Ltd ISBN: 0-470-85750-1 (HB); 0-470-85751-X (PB) A Climate Modelling Primer THIRD EDITION Kendal McGuffie University of Technology, Sydney, Australia and Ann Henderson-Sellers ANSTO Environment, Australia Copyright © 2005 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wileyeurope.com or www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to [email protected], or faxed to (+44) 1243 770620. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The Publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered.
    [Show full text]
  • A Simple Scalar Coupled Map Lattice Model for Excitable Media
    This is a repository copy of A simple scalar coupled map lattice model for excitable media. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/74667/ Monograph: Guo, Y., Zhao, Y., Coca, D. et al. (1 more author) (2010) A simple scalar coupled map lattice model for excitable media. Research Report. ACSE Research Report no. 1016 . Automatic Control and Systems Engineering, University of Sheffield Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ A Simple Scalar Coupled Map Lattice Model for Excitable Media Yuzhu Guo, Yifan Zhao, Daniel Coca, and S. A. Billings Research Report No. 1016 Department of Automatic Control and Systems Engineering The University of Sheffield Mappin Street, Sheffield, S1 3JD, UK 8 September 2010 A Simple Scalar Coupled Map Lattice Model for Excitable Media Yuzhu Guo, Yifan Zhao, Daniel Coca, and S.A.
    [Show full text]
  • Coupled Map Lattice (CML) Approach
    Different mechanisms of synchronization : Coupled Map Lattice (CML) approach – p.1 Outline Synchronization : an universal phenomena Model (Coupled maps/oscillators on Networks) – p.2 Modelling Spatio-temporal dynamics 2- and 3-dimensional fluid flow, Coupled-oscillator models, Cellular automata, Transport models, Coupled map models Model does not have direct connections with actual physical or biological systems – p.3 Modelling Spatio-temporal dynamics 2- and 3-dimensional fluid flow, Coupled-oscillator models, Cellular automata, Transport models, Coupled map models Model does not have direct connections with actual physical or biological systems General results show some universal features – p.3 We select coupled maps(chaotic), oscillators on Networks Models. – p.4 Coupled Dynamics on Networks ε xi(t +1) = f(xi(t)) + Cijg(xi(t),xj(t)) Cij Pj P Periodic orbit Synchronized chaos Travelling wave Spatio-temporal chaos – p.5 Coupled maps Models Coupled oscillators (- Kuramoto ∼ beginning of 1980’s) Coupled maps (- K. Kaneko) - K. Kaneko, ”Simulating Physics with Coupled Map Lattices —— Pattern Dynamics, Information Flow, and Thermodynamics of Spatiotemporal Chaos”, pp1-52 in Formation, Dynamics, and Statistics of Patterns ed. K. Kawasaki et.al., World. Sci. 1990” – p.6 Studying real systems with coupled maps/oscillators models: - Ott, Kurths, Strogatz, Geisel, Mikhailov, Wolf etc . From Kuramoto to Crawford .... (Physica D, Sep 2000) Chaotic Transients in Complex Networks, (Phys Rev Lett, 2004) Turbulence in oscillatory chemical reactions, (J Chem Phys, 1994) Crowd synchrony on the Millennium Bridge, (Nature, 2005) Species extinction - Amritkar and Rangarajan, Phys. Rev. Lett. (2006) – p.7 SYNCHRONIZATION First observation (Pendulum clock (1665)) - discovered in 1665 by the famous Dutch physicists, Christiaan Huygenes.
    [Show full text]
  • The Prophet of Climate Change James Lovelock Rolling Stone
    The Prophet of Climate Change: James Lovelock : Rolling Stone 4/14/10 6:53 AM Advertisement PRINTER FRIENDLY URL: http://www.rollingstone.com/politics/story/16956300/the_prophet_of_climate_change_james_lovelock Rollingstone.com Back to The Prophet of Climate Change: James Lovelock The Prophet of Climate Change: James Lovelock One of the most eminent scientists of our time says that global warming is irreversible — and that more than 6 billion people will perish by the end of the century JEFF GOODELL Posted Nov 01, 2007 2:20 PM ADVERTISEMENT At the age of eighty-eight, after four children and a long and respected career as one of the twentieth century's most influential scientists, James Lovelock has come to an unsettling conclusion: The human race is doomed. "I wish I could be more hopeful," he tells me one sunny morning as we walk through a park in Oslo, where he is giving a talk at a university. Lovelock is a small man, unfailingly polite, with white hair and round, owlish glasses. His step is jaunty, his mind lively, his manner anything but gloomy. In fact, the coming of the Four Horsemen -- war, famine, pestilence and death -- seems to perk him up. "It will be a dark time," Lovelock admits. "But for those who survive, I suspect it will be rather exciting." In Lovelock's view, the scale of the catastrophe that awaits us will soon become obvious. By 2020, droughts and other extreme weather will be commonplace. By 2040, the Sahara will be moving into Europe, and Berlin will be as hot as Baghdad.
    [Show full text]
  • Predictability: a Way to Characterize Complexity
    Predictability: a way to characterize Complexity G.Boffetta (a), M. Cencini (b,c), M. Falcioni(c), and A. Vulpiani (c) (a) Dipartimento di Fisica Generale, Universit`adi Torino, Via Pietro Giuria 1, I-10125 Torino, Italy and Istituto Nazionale Fisica della Materia, Unit`adell’Universit`adi Torino (b) Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Str. 38 , 01187 Dresden, Germany (c) Dipartimento di Fisica, Universit`adi Roma ”la Sapienza”, Piazzale Aldo Moro 5, 00185 Roma, Italy and Istituto Nazionale Fisica della Materia, Unit`adi Roma 1 Contents 1 Introduction 4 2 Two points of view 6 2.1 Dynamical systems approach 6 2.2 Information theory approach 11 2.3 Algorithmic complexity and Lyapunov Exponent 18 3 Limits of the Lyapunov exponent for predictability 21 3.1 Characterization of finite-time fluctuations 21 3.2 Renyi entropies 25 3.3 The effects of intermittency on predictability 26 3.4 Growth of non infinitesimal perturbations 28 arXiv:nlin/0101029v1 [nlin.CD] 17 Jan 2001 3.5 The ǫ-entropy 32 4 Predictability in extended systems 34 4.1 Simplified models for extended systems and the thermodynamic limit 35 4.2 Overview on the predictability problem in extended systems 37 4.3 Butterfly effect in coupled map lattices 39 4.4 Comoving and Specific Lyapunov Exponents 41 4.5 Convective chaos and spatial complexity 42 4.6 Space-time evolution of localized perturbations 46 4.7 Macroscopic chaos in Globally Coupled Maps 49 Preprint submitted to Elsevier Preprint 30 April 2018 4.8 Predictability in presence of coherent structures 51 5
    [Show full text]
  • Axel Bronstert 2005.Pdf
    I Axel Bronstert Jesus Carrera Pavel Kabat Sabine Lütkemeier Coupled Models for the Hydrological Cycle Integrating Atmosphere, Biosphere, and Pedosphere III Axel Bronstert Jesus Carrera Pavel Kabat Sabine Lütkemeier (Editors) Coupled Models for the Hydrological Cycle Integrating Atmosphere, Biosphere, and Pedosphere With 92 Figures, 19 in colour and 20 Tables IV Preface Axel Bronstert Pavel Kabat University of Potsdam Wageningen University and Institute for Geoecology Research Centre Chair for Hydrology and Climatology ALTERRA Green World Research P.O. Box 60 15 53 Droevendaalsesteeg 3 14415 Potsdam 6708 PB Wageningen Germany The Netherlands Jesus Carrera Sabine Lütkemeier Technical University of Catalonia (UPC) Potsdam Institute for Department of Geotechnical Engineering Climate Impact Research and Geosciences Telegrafenberg School of Civil Engineering 14473 Potsdam Campus Nord, Edif. D-2 Germany 08034 Barcelona Spain Library of Congress Control Number: 2004108318 ISBN 3-540-22371-1 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifi cally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfi lm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc.
    [Show full text]
  • A Robust Image-Based Cryptology Scheme Based on Cellular Non-Linear Network and Local Image Descriptors
    PRE-PRINT IJPEDS A robust image-based cryptology scheme based on cellular non-linear network and local image descriptors Mohammad Mahdi Dehshibia, Jamshid Shanbehzadehb, and Mir Mohsen Pedramb aDepartment of Computer Engineering, Faculty of Computer and Electrical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran; bDepartment of Electrical and Computer Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran ARTICLE HISTORY Compiled August 14, 2018 ABSTRACT Cellular nonlinear network (CNN) provides an infrastructure for Cellular Automata to have not only an initial state but an input which has a local memory in each cell with much more complexity. This property has many applications which we have investigated it in proposing a robust cryptology scheme. This scheme consists of a cryptography and steganography sub-module in which a 3D CNN is designed to produce a chaotic map as the kernel of the system to preserve confidentiality and data integrity in cryptology. Our contributions are three-fold including (1) a feature descriptor is applied to the cover image to form the secret key while conventional methods use a predefined key, (2) a 3D CNN is used to make a chaotic map for making cipher from the visual message, and (3) the proposed CNN is also used to make a dynamic k-LSB steganography. Conducted experiments on 25 standard images prove the effectiveness of the proposed cryptology scheme in terms of security, visual, and complexity analysis. KEYWORDS Cryptography; Steganography; Chaotic map; Cellular Non-linear Network; Image Descriptor; Complexity 1. Introduction Sensitive and personal data are parts of data transmission over the Internet which have always been suspicious to be intercepted.
    [Show full text]
  • SPATIOTEMPORAL CHAOS in COUPLED MAP LATTICE Itishree
    SPATIOTEMPORAL CHAOS IN COUPLED MAP LATTICE By Itishree Priyadarshini Under the Guidance of Prof. Biplab Ganguli Department of Physics National Institute of Technology, Rourkela CERTIFICATE This is to certify that the project thesis entitled ” Spatiotemporal chaos in Cou- pled Map Lattice ” being submitted by Itishree Priyadarshini in partial fulfilment to the requirement of the one year project course (PH 592) of MSc Degree in physics of National Institute of Technology, Rourkela has been carried out under my super- vision. The result incorporated in the thesis has been produced by developing her own computer codes. Prof. Biplab Ganguli Dept. of Physics National Institute of Technology Rourkela - 769008 1 ACKNOWLEDGEMENT I would like to acknowledge my guide Prof. Biplab Ganguli for his help and guidance in the completion of my one-year project and also for his enormous moti- vation and encouragement. I am also very much thankful to research scholars whose encouragement and support helped me to complete my project. 2 ABSTRACT The sensitive dependence on initial condition, which is the essential feature of chaos is demonstrated through simple Lorenz model. Period doubling route to chaos is shown by analysis of Logistic map and other different route to chaos is discussed. Coupled map lattices are investigated as a model for spatio-temporal chaos. Diffusively coupled logistic lattice is studied which shows different pattern in accordance with the coupling constant and the non-linear parameter i.e. frozen random pattern, pattern selection with suppression of chaos , Brownian motion of the space defect, intermittent collapse, soliton turbulence and travelling waves. 3 Contents 1 Introduction 3 2 Chaos 3 3 Lorenz System 4 4 Route to Chaos 6 4.1 PeriodDoubling.............................
    [Show full text]
  • Analysis of the Coupled Logistic Map
    Theoretical Population Biology TP1365 Theoretical Population Biology 54, 1137 (1998) Article No. TP981365 Spatial Structure, Environmental Heterogeneity, and Population Dynamics: Analysis of the Coupled Logistic Map Bruce E. Kendall* Department of Ecology and Evolutionary Biology, University of Arizona, Tucson, Arizona 85721, and National Center for Ecological Analysis and Synthesis, University of California, Santa Barbara, California 93106 and Gordon A. Fox- Department of Biology 0116, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093-0116, and Department of Biology, San Diego State University, San Diego, California 92182 Spatial extent can have two important consequences for population dynamics: It can generate spatial structure, in which individuals interact more intensely with neighbors than with more distant conspecifics, and it allows for environmental heterogeneity, in which habitat quality varies spatially. Studies of these features are difficult to interpret because the models are complex and sometimes idiosyncratic. Here we analyze one of the simplest possible spatial population models, to understand the mathematical basis for the observed patterns: two patches coupled by dispersal, with dynamics in each patch governed by the logistic map. With suitable choices of parameters, this model can represent spatial structure, environmental heterogeneity, or both in combination. We synthesize previous work and new analyses on this model, with two goals: to provide a comprehensive baseline to aid our understanding of more complex spatial models, and to generate predictions about the effects of spatial structure and environmental heterogeneity on population dynamics. Spatial structure alone can generate positive, negative, or zero spatial correlations between patches when dispersal rates are high, medium, or low relative to the complexity of the local dynamics.
    [Show full text]
  • The Dynamic Response of the Global Atmosphere-Vegetation Coupled System
    THE UNIVERSITY OF READING Department of Meteorology The Dynamic Response of the Global Atmosphere-Vegetation coupled system John K. Hughes A thesis submitted for the degree of Doctor of Philosophy October 2003 ’Declaration I confirm that this is my own work and the use of all material from other sources has been properly and fully acknowledged’ John Hughes The Dynamic Response of the Atmosphere-Vegetation coupled system Abstract Concern about future changes in the carbon cycle have highlighted the importance of a dy- namic representation of the carbon cycle in models, yet this has not been fully assessed. In this thesis, we investigate the dynamic carbon cycle model included within the Hadley Centre’s model. In order to understand the behaviour of the vegetation model a simplified model, describing the behaviour of a single plant functional type is derived. The ability of the simplified model to simulate vegetation dynamics is validated against the behaviour of the full complexity model. The dynamical properties of the simplified model are then investigated. To further investigate the dynamic response of vegetation, a 300 year climate model simulation of terrestrial vegetation re-growth from global desert has been performed. Vegetation is shown to introduce large time lags in land surface properties. This large memory in the terrestrial carbon cycle is an important result for GCM simulations. The large timescale also affects the response of existing vegetation to climatic forcings. The behaviour of the land surface in terms of source-sink transitions of atmospheric CO2 is discussed. It is found that the transitions between source and sink of CO2 are dependant on the vegetation timescales.
    [Show full text]