MATHEMATICAL MODELS IN POPULATION DYNAMICS BY ALEXANDER SALISBURY A Thesis Submitted to the Division of Natural Sciences New College of Florida in partial fulfillment of the requirements for the degree Bachelor of Arts Under the sponsorship of Dr. Necmettin Yildirim Sarasota, FL April, 2011 ii ACKNOWLEDGEMENTS I would like to thank my advisor Dr. Necmettin Yildirim for his support, guidance, and seemingly unlimited supply of patience. Additional thanks to my thesis committee members Dr. Chris Hart and Dr. Eirini Poimenidou for their guidance and criticism. Final thanks to family and friends for their love and support. iii TABLE OF CONTENTS Acknowledgements .............................................................................................................................. ii Table of Contents.................................................................................................................................. iii List of Tables and Figures ................................................................................................................. vi Abstract .....................................................................................................................................................1 Chapter 1: Background ........................................................................................................................2 1.1 What are Dynamical Systems? ............................................................................................................. 2 1.2 Formulating the Model ........................................................................................................................... 5 1.3 Methods for Analysis of Population Dynamics .............................................................................. 8 Solving Differential Equations ................................................................................................................ 8 Expressing in Dimensionless Form ...................................................................................................... 8 One-Dimensional Models: Geometrical Analysis ......................................................................... 10 One-Dimensional Models: Local Linearization ............................................................................. 11 Two-Dimensional Models: Geometrical Analysis ........................................................................ 13 Two-Dimensional Models: Local Linearization ............................................................................ 15 Classification of Equilibria .................................................................................................................... 19 1.4 An Historical Overview of Population Dynamics....................................................................... 23 Fibonacci ...................................................................................................................................................... 25 Leonhard Euler .......................................................................................................................................... 26 Daniel Bernoulli ........................................................................................................................................ 27 Thomas Robert Malthus ......................................................................................................................... 28 Pierre-François Verhulst ....................................................................................................................... 29 Leland Ossian Howard and William Fuller Fiske ......................................................................... 32 Raymond Pearl .......................................................................................................................................... 33 iv Alfred James Lotka and Vito Volterra ............................................................................................... 36 Anderson Gray McKendrick and William Ogilvy Kermack ....................................................... 40 Georgy Frantsevich Gause ..................................................................................................................... 43 Chapter 2: Single-Species Population Models ........................................................................... 45 2.1 Malthusian Exponential Growth Model ......................................................................................... 47 Analytic Solution ....................................................................................................................................... 47 Geometrical Analysis ............................................................................................................................... 48 Assumptions of the Model ..................................................................................................................... 50 2.2 Classical Logistic Growth Model ...................................................................................................... 51 Analytic Solution ....................................................................................................................................... 52 Obtaining Equilibrium Points .............................................................................................................. 53 Geometrical Analysis ............................................................................................................................... 54 Local Linearization .................................................................................................................................. 56 Assumptions of the Model ..................................................................................................................... 57 2.3 Theta Logistic Growth Model ............................................................................................................ 58 2.4 Logistic Model with Allee Effect ....................................................................................................... 61 Geometrical Analysis ............................................................................................................................... 63 2.5 Growth Model with Multiple Equilibria ........................................................................................ 65 Geometrical Analysis ............................................................................................................................... 66 Chapter 3: Multispecies Population Models .............................................................................. 68 3.1 Interspecific Competition Model...................................................................................................... 71 Obtaining Equilibrium Points .............................................................................................................. 72 Geometrical Analysis ............................................................................................................................... 73 Local Linearization .................................................................................................................................. 80 3.2 Facultative Mutualism Model ............................................................................................................ 82 v Obtaining Equilibrium Points .............................................................................................................. 83 Geometrical Analysis ............................................................................................................................... 84 Local Linearization .................................................................................................................................. 86 3.3 Obligate Mutualism Model ................................................................................................................. 88 Geometrical Analysis ............................................................................................................................... 88 3.4 Predator-Prey Model ............................................................................................................................ 92 Geometrical Analysis ............................................................................................................................... 93 Local Linearization .................................................................................................................................. 96 Chapter 4: Concluding Remarks .................................................................................................... 98 References .......................................................................................................................................... 101 vi LIST OF TABLES AND FIGURES Figure 1.1 .........................................................................................................................................4 Figure 1.2 .........................................................................................................................................5 Figure 1.3 .......................................................................................................................................11 Figure 1.4 .......................................................................................................................................15