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Other___________________________________________________________________________________ University Microfilms International DIFFUSION ANALYSIS AND STATIONARY DISTRIBUTION OF THE LOTTERY COMPETITION MODEL DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University by Jeffrey Scott Hatfield, B.A., M.S., M.A.S. ***** Tin Ohio State University 1986 Dissertation Committee: Approved by P.L. Chesson J.D. Powers __________ Adviser R.C. Srivastava Department of Statistics Copyright by Jeffrey Scott Hatfield 1986 ACKNOWLEDGMENTS I would like to thank my graduate committee for their suggestions in the preparation of this document. In particular. I am grateful to my adviser, Peter L. Chesson, for his guidance during my graduate education. My deepest appreciation goes to my parents for their support throughout my education. Finally, I would like to give a special thanks to Bryan Knedler for his friendship and encouragement while I was in pursuit of this degree. VITA October 4, 1957 ........................................................... .Bom, Cincinnati, Ohio 1979 ........................................................................... B .A , Miami University. Oxford, Ohio 1979 - 1981 ................................................................ Teaching Assistant, Miami University, Department of Zoology, Oxford, Ohio 1981 ........................................................................... M.S., Miami University, Oxford, Ohio 1981 - 1982 ................................................................ University Fellowship, The Ohio State University, Department of Statistics, Columbus, Ohio 1982 ........................................................................... M. A.S., The Ohio State University, Columbus, Ohio 1982 - 1985 ................................................................ Teaching Assistant, The Ohio State University, Department of Statistics, Columbus, Ohio 1986 ........................................................................... Presidential Fellowship, The Ohio State University, Department of Statistics, Columbus, Ohio PUBLICATIONS Hatfield, J.S., T.E. Wissing, S.I. Guttman, and M.P. Farrell. 1982. Electrophoretic analysis of gizzard shad from the lower Mississippi River and Ohio, Trans. Am. Fish. Soc. I l l , 742-748. FIELDS OF STUDY Major Field: Biostatistics iii TABLE OF CONTENTS ACKNOWLEDGMENTS..................................................................................................ii VTTA.................................. ......................................................................................... iii LIST OF FIGURES......................................................................................................... vi INTRODUCTION............................................................................................................. 1 CHAPTER PAGE I. DIFFUSION ANALYSIS AND STATIONARY DISTRIBUTION OF THE TWO-SPECIES LOTTERY COMPETITION MODEL...................................5 Introduction. ........................................................................................ 5 The Stationary Distribution. ................................................................. 8 Species Diversity Indices .....................................................................11 Comparison with other Models ............................................................ 11 Conclusion. .........................................................................................16 II. THE MULTISPECIES LOTTERY COMPETITION MODEL: A DIFFUSION ANALYSIS...................................................................................................18 Introduction ...................................................... 18 The Stationary Distribution ................................................................. 21 Species Diversity Indices .................................................................... 23 Conclusion. ........................................................................................ 24 III. DIFFUSION ANALYSIS AND STATIONARY DISTRIBUTION OF THE TWO-SPECIES LOTTERY COMPETITION MODEL WITH JUVENILE PRESETTLEMENT COMPETITION...........................................................26 Introduction ........................................................................................ 26 The Stationary Distribution ................................................................. 29 Conclusion .........................................................................................30 General Conclusion. ........................................................................... 31 APPENDICES A. DERIVATIONS FOR CHAPTER 1................................................................35 Diffusion Coefficients .........................................................................35 iv Boundary Classification ...................................................................... 36 Comparison of Coexistence Criteria. .................................................. 40 B. DERIVATIONS FOR CHAPTER II.............................................................. 42 Diffusion Coefficients ........................................................................ 42 C. DERIVATIONS FOR CHAPTER III............................................................. 44 Diffusion Coefficients .........................................................................44 Boundary Classification ......................................................................47 LIST OF REFERENCES................................................................................................ 50 v LIST OF FIGURES FIGURES PAGE 1. Comparison of the coexistence conditions of equation (1.9) (dashed lines) with the numerical calculations of Chesson and Warner (1981) (solid lines) for different values of the geometric mean death rate, d. Note that for d = 0 both conditions are identical. The coexistence region for a given d is the area above the line specified by that particular d ..................................................... 12 INTRODUCTION Competition among two or more species of organisms has been demonstrated time and again fee vsrious species of animals and plants (e.g. amphibians - Hairston, 1981; Morin, 1983; 1986; barnacles-Connell, 1961a; 1961b; birds-Gram, 1968; Dhoarifc, 1977; Dhondt and Eyckerman, 1980; Garcia, 1983; Grant and Schluter, 1984; fish-H ixon, 1980; Fauscle and White, 1981; Werner, 1984; insects-Park, 1962;Rathcke, 1976; McAuliffe, 1984; plants-Handel. 1978; Lobchenco, 1978; 1986; Platt and Weis, 198S; reptiles - Pacala and Rooghgarden, 1982; Salzbutg, 1984; rodents - Sheppard, 1971; Gram, 1972; Brown, 1975; Glass and Slade, 1980; Frye, 1983; and spiders-Spiller, 1984; Wise, 1984). Reviews by Connell (1983) and Schoener ( 1983)offield experiments attempting to find evidenceof interspecific