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When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given e.g. AUTHOR (year of submission) "Full thesis title", University of Southampton, name of the University School or Department, PhD Thesis, pagination http://eprints.soton.ac.uk THE DAISYWORLD CONTROL SYSTEM James Dyke Submitted for the degree of DPhil University of Sussex October 2009 Summary The original Gaia Hypothesis proposed that life on Earth, along with the oceans, at- mosphere and crust, forms a homeostatic system which reduces the effects of exter- nal perturbations, so that conditions are maintained to within the range that allows widespread life. Daisyworld is a simple mathematical model intended to demonstrate certain aspects of this planetary homeostasis. There have been a considerable number of extensions and developments to the original Daisyworld model. Some of this work has been produced in response to criticism of the Gaia Hypothesis and Daisyworld specifically and some has been produced by using Daisyworld as a testbed to explore a range of issues. This thesis examines the Daisyworld control system and in doing so explains how Daisyworld performs homeostasis. The control system is classified as a rein control sys- tem which is potentially applicable to a wide range of scenarios from physiological and environmental homeostasis to robotic control. A series of simple Daisyworld models are produced and aspects of the original Daisyworld are explained, in particular the inverse response to forcing: why temperature goes down on Daisyworld when the brightness of the star increases. The Daisyworld control system is evaluated within an evolutionary context. A key result is that environmental regulation emerges not despite of Darwinian evolution but because of it. Within an ecological context, it is found that increasing the complexity of a self-regulating ecosystem can increase its stability. An energy balance climate model is developed to assess the effects of non-equilibrium thermodynamic processes on the Daisyworld control system. Results are presented that support the hypothesis that when the system is in a state of maximum entropy production, homeostasis is maximised. ii Declaration I hereby declare that this thesis has not been submitted, either in the same or different form, to this or any other University for a degree. Signature: iii Acknowledgments First, I would like to recognise the contributions of my doctoral supervisor Inman Harvey to this thesis. The time spent with tea and tiffin cake while working on the white board were some of the most productive and happiest moments of my research. Tim Lenton, who acted as an unofficial supervisor was a constant source of support and encouragement and introduced me to the wonders of the Bengal Spice. I would like to thank Axel Kleidon, for his understanding and support during the very final stages of this thesis. I would like to thank Jim and Sandy Lovelock for their generosity, hospitality and inspiration. Many thanks are due to those that have been kind enough to suffer my contributions to collaborative projects: Graeme Ackland, Jamie McDonald-Gibson, Ezequiel di Paolo and Hywell Williams. Of particular note is Jamie Wood. I am very appreciative of his efforts to get me a job. This thesis would not exist without Mandi Dyke's constant and at times heroic support, encouragement and ruthless proof reading and copy editing. Thank you. You continue to believe in me. You continue to amaze me. I would not exist without my parents. So thanks Mum and Dad. Further thanks are due for giving me the opportunities that reason alone would have most probably concluded would have been wasted. I was funded while doing the research for this thesis by the Engineering and Physical Sciences Research Council. I was very fortunate to have been paid to study. Finally I dedicate this thesis to Leo and Alexander. It is a natural desire to bring children into a world better than the one you were born into. I hope that the following pages may contain an idea or result or insight that may, at least, not make things any worse. iv Publications Parts of this thesis have been previously published in journal articles and conference proceedings: Dyke J. G. and Harvey I. R., (2005) Hysteresis and the Limits of Homeostasis: from Daisyworld to Phototaxis, Proceedings of VIIIth European Conference on Artificial Life, ECAL 2005, eds Capcarrere, M. and Freitas, A. and Bentley, J. and Johnson, C. and Timmis, J, pp241-246, Springer, Berlin. Dyke J. G. and I. R. Harvey, (2006) Pushing up the daisies, Proceedings of the Tenth International Conference on the Simulation and Synthesis of Living Systems, ALife X, eds L.M. Rocha and L.S. Yager and M.A. Bedau and D. Floreano and R.L. Goldstone and A. Vespignani, pp426-431, MIT Press, Cambridge MA. Dyke, J. G. and McDonald-Gibson, J. and Di Paolo, E. and Harvey, I. R., (2007) In- creasing complexity can increase stability in a self-regulating ecosystem, Proceedings of IXth European Conference on Artificial Life, ECAL 2007, pp133-142, Springer, Berlin. Wood, A. J. and Ackland, G. J. and Dyke, J. G. and Williams, H. T. P. and Lenton, T. M. (2008) Daisyworld: a review, Reviews of Geophysics 46 doi:10.1029/2006RG000217. McDonald-Gibson, J. and Dyke, J. G. and Di Paolo, E. and Harvey, I. R. (2008), Environmental Regulation Can Arise Under Minimal Assumptions, Journal of Theo- retical Biology 254(4), pp653-666. Dyke, J. G. (2008), Entropy production in an energy balance Daisyworld model, Pro- ceedings of the Eleventh International Conference on the Simulation and Synthesis of Living Systems, ALIFE'11, MIT Press, Cambridge MA. v Contents Summary ii Declaration iii Acknowledgments iv Publications v 1 Introduction 1 1.1 Why Did I Write This Thesis? . 1 1.2 Target Audience . 3 1.2.1 Earth system scientists . 3 1.2.2 Artificial Life practitioners . 3 1.2.3 Ecological and evolutionary theorists . 4 1.3 Layout of Thesis . 5 1.3.1 Chapter 2 . 5 1.3.2 Chapter 3 . 5 1.3.3 Chapter 4 . 6 1.3.4 Chapter 5 . 6 1.3.5 Chapter 6 . 6 1.3.6 Chapter 7 . 7 1.3.7 Chapter 8 . 7 1.3.8 Chapter 9 . 8 1.4 Main results of thesis . 8 1.5 Summary . 9 2 Background 10 2.1 Overview . 10 vi 2.2 Definitions and Concepts . 10 2.2.1 Homeostasis . 10 2.3 Homeorhesis . 12 2.4 Control Theory . 13 2.4.1 Fixed points and steady-state `error' . 14 2.5 Planetary Homeostasis and Gaia Theory . 15 2.6 Daisyworld . 17 2.6.1 Daisyworld Homeostasis . 18 2.7 Original Daisyworld . 18 2.7.1 Hysteresis . 23 2.8 Analytical Solutions . 23 2.9 Developments of Daisyworld . 25 2.9.1 Spatial Extensions . 26 2.9.2 Ecosystem Dynamics . 26 2.9.3 Adaptation of Albedo . 27 2.9.4 Adaptation of Optimal Growth Temperature . 27 2.9.5 Non-radiative Models . 28 2.9.6 Conceptual Models . 28 2.10 Summary . 29 3 Simplified Daisyworld 30 3.1 Overview . 30 3.2 Simplifying Daisyworld . 30 3.3 Model . 31 3.4 Analysis . 37 3.4.1 White Daisies Only . 38 3.4.2 Limits of White Daisy Growth . 42 3.4.3 Black daisies only . 42 3.4.4 Limits of Black Daisy Growth . 44 3.4.5 Black and White Daisies . 44 3.5 Investigating the Inverse Response . 50 3.5.1 What does q mean? . 52 3.5.2 The inverse response and q ..................... 54 3.6 Summary . 55 vii 4 Two Box Daisyworld 57 4.1 Previous Publications . 57 4.2 Overview . 57 4.3 Two Box Daisyworld . 58 4.4 Model . 58 4.5 Resource Competition and Zero Steady State Error . 62 4.6 Hysteresis and the Limits of Homeostasis . 65 4.7 Harvey TBDW . 65 4.7.1 Results . 67 4.8 Cable Car Model . 71 4.9 Model . 73 4.10 Results . 74 4.11 Hysteresis and the Limits of Phototaxis . 74 4.12 Analysis . 76 4.13 Summary . 77 5 Rein Control 79 5.1 Overview . 79 5.2 Unidirectional Rein Control . 80 5.3 Daisyworld Effect Relationships . 81 5.4 A Simple Rein Control Dynamical System . 83 5.5 Single Rein Controller . 85 5.5.1 Type ii System . 85 5.5.2 Type iii System . 86 5.6 Two Rein Control Systems . 89 5.6.1 Benign Interactions . 89 5.6.2 Antagonistic Interactions . 90 5.7 Daisyworld and Rein Control . 93 5.8 Limits of Regulation . 93 5.9 `Lucky Gaia' . 94 5.10 Summary . 95 6 Evolving Rein Control 96 6.1 Previous Publications . 96 6.2 Overview . 96 viii 6.3 Reduced Assumption Daisyworld . 97 6.4 Model . 99 6.4.1 Algorithm . ..