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Views by Connell (1983) and Schoener INFORMATION TO USERS While the most advanced technology has been used to photograph and reproduce this manuscript, the quality of the reproduction is heavily dependent upon the quality of the material submitted. For example: ® Manuscript pages may have indistinct print. In such cases, the best available copy has been filmed. • Manuscripts may not always be complete. In such cases, a note will indicate that it is not possible to obtain missing pages. • Copyrighted material may have been removed from the manuscript. In such cases, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, and charts) are photographed by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each oversize page is also filmed as one exposure and is available, for an additional charge, as a standard 35mm slide or as a 17”x 23” black and white photographic print. Most photographs reproduce acceptably on positive microfilm or microfiche but lack the clarity on xerographic copies made from the microfilm. For an additional charge, 35mm slides of 6”x 9” black and white photographic prints are available for any photographs or illustrations that cannot be reproduced satisfactorily by xerography. 8703557 Hatfield, Jeffrey Scott DIFFUSION ANALYSIS AND STATIONARY DISTRIBUTION OF THE LOTTERY COMPETITION MODEL The Ohio State University Ph.D. 1986 University Microfilms International300 N. Zeeb Road, Ann Arbor, Ml 48106 Copyright 1986 by Hatfield, Jeffrey Scott All Rights Reserved PLEASE NOTE: In all cases this material has been filmed in the best possible way from the available copy. Problems encountered with this document have been identified here with a check mark V . 1. Glossy photographs or pages ______ 2. Colored illustrations, paper or print ________ 3. Photographs with dark background_____ 4. Illustrations are poor co______ py 5. Pages with black marks, not original copy _______ 6. Print shows through as there is text on both sides of page _______ 7. Indistinct, broken or small print on several pages \/ 8. Print e x c e e d s margin req u irem en ts_______ 9. Tightly bound copy with print lost in spine ________ 10. Computer printout pages with indistinct print _______ 11. Page(s) ______________ lacking when material received, and not available from school or author. 12. Page(s) ______________ seem to be missing in numbering only as text follows. 13. Two pages numbered . Text follows. 14. Curling and wrinkled pages _______ 15. Dissertation contains pages with print at a slant, filmed as _________received 16. 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
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