Lecture Notes in Biomathematics – serie
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Häftad, Engelska, 1989
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Depth Perception in Frogs and Toads provides a comprehensive exploration of the phenomenon of depth perception in frogs and toads, as seen from a neuro-computational point of view. Perhaps the most important feature of the book is the development and presentation of two neurally realizable depth perception algorithms that utilize both monocular and binocular depth cues in a cooperative fashion. One of these algorithms is specialized for computation of depth maps for navigation, and the other for the selection and localization of a single prey for prey catching. The book is also unique in that it thoroughly reviews the known neuroanatomical, neurophysiological and behavioral data, and then synthesizes, organizes and interprets that information to explain a complex sensory-motor task. The book will be of special interest to that segment of the neural computing community interested in understanding natural neurocomputational structures, particularly to those working in perception and sensory-motor coordination. It will also be of interest to neuroscientists interested in exploring the complex interactions between the neural substrates that underly perception and behavior.
Del 1 - Lecture Notes in Biomathematics
Deterministic Threshold Models in the Theory of Epidemics
Häftad, Engelska, 1974
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These notes correspond to a set of lectures given at the Univer sity of Alberta during the spring semester, 1973. The first four sec tions present a systematic development of a deterministic, threshold model for the spraad of an infection. Section 5 presents some compu tational results and attempts to tie the model with other mathematics. In each of the last three sections a separate, specialized topic is presented. The author wishes to thank Professor F. Hoppensteadt for making available preprints of two of his papers and for reading and comment ing on a preliminary version of these notes. He also wishes to thank Professor J. Mosevich for providing the graphs in Section 5. The visit at the University of Alberta was a very pleasant one and the author wishes to express his appreciation to Professors S. Ghurye and J. Macki for the invitation to visit there. Finally, thanks are due to the very competent secretarial staff at the University of Alberta for typing the original draft of the lecture notes and to Mrs. Ada Burns of the University of Iowa for her excellent typescript of the final version. TABLE OF CONTENTS 1. A Simple Epidemic Model with Permanent Removal . . . • . . . 1 2. A More General Model and the Determination of the Intensity of an Epidemic. 10 21 3. A Threshold Model. 4. A Threshold Model with Temporary Immunity. 34 5. Some Special Cases and Some Numerical Examples 48 A Two Population Threshold Model . 62 6.
Del 2 - Lecture Notes in Biomathematics
Mathematical Problems in Biology
Victoria Conference
Häftad, Engelska, 1974
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A conference on "Some Mathematical Problems in Biology" was held at the University of Victoria, Victoria, B. C. , Canada, from May 7 - 10, 1973. The participants and invited speakers were mathematicians interested in problems of a biological nature, and scientists actively engaged in developing mathematical models in biological fields. One aim of the conference was to attempt to assess what the recent rapid growth of mathematical interaction with the biosciences has accomplished and may accomplish in the near future. The conference also aimed to expose the problems of communication bet~",een mathematicians and biological scientists, and in doing so to stimulate the interchange of ideas. It was recognised that the topic spans an enormous breadth, and little attempt was made to balance the very diverse areas. Widespread active interest was shown in the conference, and just over one hundred people registered. The varied departments and institutions across North America from which the participants came made it both academically and geographically mixed. The chief activity of the conference was the presentation of papers. Nine invited guest speakers (see table of contents) each gave a one hour talk. These covered a wide range of topics. There were twenty-five shorter (twenty minute) contributed papers, and almost all papers l,rere followed by a five minute question and discussion period. Duplicated abstracts of presented papers were available at the meeting. An evening informal discussion meeting of participants, chaired by Dr. A. B. Tayler, and led by Drs. E. M. Hagmeier, E. C.
Häftad, Engelska, 1974
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These notes serve as an introduction to stochastic theories which are useful in population biology; they are based on a course given at the Courant Institute, New York, in the Spring of 1974. In order to make the material. accessible to a wide audience, it is assumed that the reader has only a slight acquaintance with probability theory and differential equations. The more sophisticated topics, such as the qualitative behavior of nonlinear models, are approached through a succession of simpler problems. Emphasis is placed upon intuitive interpretations, rather than upon formal proofs. In most cases, the reader is referred elsewhere for a rigorous development. On the other hand, an attempt has been made to treat simple, useful models in some detail. Thus these notes complement the existing mathematical literature, and there appears to be little duplication of existing works. The authors are indebted to Miss Jeanette Figueroa for her beautiful and speedy typing of this work. The research was supported by the National Science Foundation under Grant No. GP-32996X3. CONTENTS I. LINEAR MODELS •••••. •••••••••••••••. . ••••••••••••••••••••••••••••••••••••••• 1 1. The Poisson Process ••••••••••••••••••••••••••. ••. •••••••••••. •••••••••• 1 2. Birth and Death Processes 5 2. 1 Linear Birth Process 5 2. 2 Linear Birth and Death Process •••••. ••••••. ••••••••••••••. •••••••• 7 2. 3 Birth and Death with Carrying Capacity ••••••••. •••. ••••••. ••••••. • 163. Branching Processes •••••••••••••••••••. •••••••. ••••••••. ••••••••••••••• 20 3. 1 Continuous Time . •••. ••••••••••. •••••••••••••••••••. ••••••••. ••••.
Del 4 - Lecture Notes in Biomathematics
Physics and Mathematics of the Nervous System
Proceedings of a Summer School organized by the International Centre for Theoretical Physics, Trieste, and the Institute for Information Sciences, University of Tübingen, held at Trieste, August 21–31, 1973
Häftad, Engelska, 1974
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This volume is the record and product of the Summer School on the Physics and Mathematics of the Nervous System, held at the International Centre for Theoretical Physics in Trieste from August 21-31, 1973, and jointly organized by the Institute for Information Sciences, University of Tlibingen and by the Centre. The school served to bring biologists, physicists and mathemati cians together to exchange ideas about the nervous system and brain, and also to introduce young scientists to the field. The program, attended by more than a hundred scientists, was interdisciplinary both in character and participation. The primary support for the school was provided by the Volkswagen Foundation of West Germany. We are particularly indebted to Drs. G. Gambke, M. -L Zarnitz, and H. Penschuck of the Foundation for their in terest in and help with the project. The school also received major support from the International Centre for Theoretical Physics in Trieste and its sponsoring agencies, including the use of its excellent facili ties. We are deeply indebted to Professor A. Salam for his kind co operation and also to Professor P. Budini, Dr. A. M. Hamende, and to the many members of the Centre staff whose hospitality and efficiency con tributed so much to the success of the school. We are pleased to acknow ledge the generous ~id and cooperation of the University of Tlibingen and would like to thank its President, A.
Del 5 - Lecture Notes in Biomathematics
Mathematical Analysis of Decision Problems in Ecology
Proceedings of the NATO Conference held in Istanbul, Turkey, July 9–13, 1973
Häftad, Engelska, 1975
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The contents of this volume involve selection, emendation and up-dating of papers presented at the NATO Conference "Mathe matical Analysis of Decision problems in Ecology" in Istanbul, Turkey, July 9-13, 1973. It was sponsored by the System Sciences Division of NATO directed by Dr. B. Bayraktar with local arrange ments administered by Dr. Ilhami Karayalcin, professor of the Department of Industrial Engineering at the Technical University of Istanbul. It was organized by A. Charnes, University professor across the University of Texas System, and Walter R.Lynn, Di rector of the School of Civil and Environmental Engineering at Cornell Unjversity. The objective of the conference was to bring together a group of leading researchers from the major sciences involved in eco logical problems and to present the current state of progress in research of a mathematical nature which might assist in the solu tion of these problems. Although their presentations are not herein recorded, the key note address of Dr. John A. Logan, president, Rose-Hulman Institute of Technology and the presentation of Dr. Michael J. L. Kirby, then Assistant principal Secretary to the premier of Nova Scotia, Canada, greatly influenced the discussion of papers and their subsequent emendations and revisions.
Del 6 - Lecture Notes in Biomathematics
Modeling and Control in the Biomedical Sciences
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These notes are based on (i) a series of lectures that I gave at the 14th Biennial Seminar of the Canadian Mathematical Congress held at the University of Western Ontario August 12-24, 1973 and (li) some of my lectures in a modeling course that I have cotaught in the Division of Bio-Medical Sciences at Brown during the past several years. An earlier version of these notes appeared in the Center for Dynamical Systems Lectures Notes series (CDS LN 73-1, November 1973). I have in this revised and extended version of those earlier notes incorporated a number of changes based both on classroom experience and on my research efforts with several colleagues during the intervening period. The narrow viewpoint of the present notes (use of optimization and control theory in biomedical problems) reflects more the scope of the CMC lectures given in August, 1973 than the scope of my own interests. Indeed, my real interests have included the modeling process itself as well as the contributions made by investiga tors who employ the techniques and ideas of control theory, systems analysis, dif ferential equations, and stochastic processes. Some of these contributions have quite naturally involved application of optimal control theory. But in my opinion many of the interesting efforts being made in modeling in the biomedical sciences encompass much more than the use of control theory.
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This book illustrates some of the ways physics and mathematics have been, and are being, used to elucidate the underlying mechan isms of passive ion movement through biological membranes in general, and the membranes of excltable cells in particular. I have made no effort to be comprehensive in my introduction of biological material and the reader interested in a brief account of single cell electro physlology from a physically-oriented biologists viewpoint will find the chapters by Woodbury (1965) an excellent introduction. Part I is introductory in nature, exploring the basic electrical properties of inexcitable and excitable cell plasma membranes. Cable theory is utilized to illustrate the function of the non-decrementing action potential as a signaling mechanism for the long range trans mission of information in the nervous system, and to gain some in sight into the gross behaviour of neurons. The detailed analysis of Hodgkin and Huxley on the squid giant axon membrane ionic conductance properties is reviewed briefly, and some facets of membrane behaviour that have been revealed since the appearance of their work are dis cussed. Part II examines the foundations of electrodiffusion theory, and the use of that theory in trying to develop quantitative expla nationsof the observed membrane properties of excitable cells, in particular the squid giant axon. In addition, an ad hoc formulation of electrodiffusion theory including active transport is presented to illustrate the qualitative nature of cellular homeostasis with respect to intracellular ionic concentrations and membrane potential, and cellular responses to prolonged stimUlation.
Del 8 - Lecture Notes in Biomathematics
Antigen Antibody Interactions
Häftad, Engelska, 1976
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1. 1 Organization of the Immune System One of the most important survival mechanisms of vertebrates is their ability to recognize and respond to the onslaught of pathogenic microbes to which they are conti- ously exposed. The collection of host cells and molecules involved in this recognition 12 response function constitutes its immune system. In man, it comprises about 10 cells 20 (lymphocytes) and 10 molecules (immunoglobulins). Its ontogenic development is c- strained by the requirement that it be capable of responding to an almost limitless variety of molecular configurations on foreign substances, while simultaneously remaining inert to those on self components. It has thus evolved to discriminate, with exquisite precision, between molecular patterns. The foreign substances which induce a response, called antigens, are typically large molecules such as proteins and polysaccharides. The portions of these with which immunoglobulins interact are called epitopes or determinants. A typical protein epitope may consist of a configuration formed by the spatial arrangements of four or five amino acids and have an average linear dimension of about 20 A.
Del 9 - Lecture Notes in Biomathematics
Stochastic Model for Immunological Feedback in Carcinogenesis: Analysis and Approximations
Häftad, Engelska, 1976
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Stochastic processes often pose the difficulty that, as soon as a model devi ates from the simplest kinds of assumptions, the differential equations obtained for the density and the generating functions become mathematically formidable. Worse still, one is very often led to equations which have no known solution and don't yield to standard analytical methods for differential equations. In the model considered here, one for tumor growth with an immunological re sponse from the normal tissue, a nonlinear term in the transition probability for the death of a tumor cell leads to the above-mentioned complications. Despite the mathematical disadvantages of this nonlinearity, we are able to consider a more sophisticated model biologically. Ultimately, in order to achieve a more realistic representation of a complicated phenomenon, it is necessary to examine mechanisms which allow the model to deviate from the more mathematically tractable linear format. Thus far, stochastic models for tumor growth have almost exclusively considered linear transition probabilities.
Del 10 - Lecture Notes in Biomathematics
Belousov-Zhabotinskii Reaction
Häftad, Engelska, 1976
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In 1958 B. P. Belousov discovered that the oxidation of citric acid by bromate in the presence of cerium ions does not proceed to equilibrium methodically and uniformly, like most chemical reactions, but rather oscillates with clocklike precision between a yellow and colorless state. See Fig. 11. 1, p. 30. A. M. Zhabotinskii followed up on Belousov's original observation and in 1964 his first investigations appeared in the Russian journal Biofizika. Though H. Degn (in Copenhagen at the time) knew of Zhabotinskii's work and published his own account of the mechanism of oscillation in Nature (1967), this interesting reaction attracted little attention among Western scientists until 1968, when Zhabotinskii and his coworkers and Busse (from Braunschweig, W. Germany) reported on their work at an international conference on biological and biochemical oscillators held in Prague. Shortly thereafter appeared· a flurry of papers on temporal oscillations and spatial patterns in this reaction system. Vavilin and Zhabotinskii (1969) [and later Kasperek and Bruice (1971)] studied the kinetics of the oxidation 3 of Ce+ by Br0 and the oxidation of organic species by Ce+4. Busse (1969) 3 reported his observation of colored bands of chemical activity propagating up and down in a long tube of unstirred solution. Zaikin and Zhabotinskii (1970) observed circular chemical waves in thin layers of solution.
Del 11 - Lecture Notes in Biomathematics
Mathematical Models in Medicine
Workshop, Mainz, March 1976
Häftad, Engelska, 1976
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On March 8/9, 1976 a workshop on "Mathematical Models in Medicine" was held at Mainz (German Federal Republic) by the group of "Mathe- matical Models" of the Deutsche Gesellschaft fUr Medizinische Doku- mentation, Informatik und Statistik. Purpose of this conference was to bring together experts from the GFR and neighbouring countries working in this field to evaluate possibilities and limits of this area of research in discussions with interested participants. This issue of Lecture Notes contains the invited contributions as well as the relevan~ remarks made by the discussants. Corresponding to the aims of the workshop the contributors had been encouraged to demon- strate their mathematical models in the light of actual applied examples. It had been our intention to restrict attention to a small number of specific areas in order to achieve a concentrated in depth treatment in these restricted areas. The areas chosen contain two - Epidemio- logy and Cell Models - which in the organisers feeling are not yet as well established in Continental Europe and one - Pharmacokinetic- with a more direct appeal to applied workers.While in the Epidemio- logy of infectious and parasitic diseases today strategies of control and eradication are gaining importance, the cell models are concerned with explaining the modes of genesis of cancerous growth and the kinetics and interactions within multi-cell structures.
Del 12 - Lecture Notes in Biomathematics
Models of the Stochastic Activity of Neurones
Häftad, Engelska, 1976
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These notes have grown from a series of seminars given at Leeds between 1972 and 1975. They represent an attempt to gather together the different kinds of model which have been proposed to account for the stochastic activity of neurones, and to provide an introduction to this area of mathematical biology. A striking feature of the electrical activity of the nervous system is that it appears stochastic: this is apparent at all levels of recording, ranging from intracellular recordings to the electroencephalogram. The chapters start with fluctuations in membrane potential, proceed through single unit and synaptic activity and end with the behaviour of large aggregates of neurones: L have chgaen this seque~~e\/~~';uggest that the interesting behaviourr~f :the nervous system - its individuality, variability and dynamic forms - may in part result from the stochastic behaviour of its components. I would like to thank Dr. Julio Rubio for reading and commenting on the drafts, Mrs. Doris Beighton for producing the final typescript and Mr. Peter Hargreaves for preparing the figures.
Del 13 - Lecture Notes in Biomathematics
Mathematical Models in Biological Discovery
Häftad, Engelska, 1977
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When I was asked to help organize an American Association for the Advancement of Science symposium about how mathematical models have con tributed to biology, I agreed immediately. The subject is of immense importance and wide-spread interest. However, too often it is discussed in biologically sterile environments by "mutual admiration society" groups of "theoreticians", many of whom have never seen, and most of whom have never done, an original scientific experiment with the biolog ical materials they attempt to describe in abstract (and often prejudiced) terms. The opportunity to address the topic during an annual meeting of the AAAS was irresistable. In order to try to maintain the integrity ;,f the original intent of the symposium, it was entitled, "Contributions of Mathematical Models to Biological Discovery". This symposium was organized by Daniel Solomon and myself, held during the 141st annual meeting of the AAAS in New York during January, 1975, sponsored by sections G and N (Biological and Medical Sciences) of the AAAS and the North American Regions of the Biometric Society, and supported by grant BMS 75-0280) from the National Science Foundation. What follows in this volume are papers by nine of the participants who not only felt that they had something to say in a symposium entitled, "Contributions of Mathematical Models to Biological Discovery", but who falso were willing to record their ideas in more detail here.
Del 14 - Lecture Notes in Biomathematics
Diffusion Processes and Related Topics in Biology
Häftad, Engelska, 1977
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These notes are based on a one-quarter course given at the Department of Biophysics and Theoretical Biology of the University of Chicago in 1916. The course was directed to graduate students in the Division of Biological Sciences with interests in population biology and neurobiology. Only a slight acquaintance with probability and differential equations is required of the reader. Exercises are interwoven with the text to encourage the reader to play a more active role and thus facilitate his digestion of the material. One aim of these notes is to provide a heuristic approach, using as little mathematics as possible, to certain aspects of the theory of stochastic processes that are being increasingly employed in some of the population biol ogy and neurobiology literature. While the subject may be classical, the nov elty here lies in the approach and point of view, particularly in the applica tions such as the approach to the neuronal firing problem and its related dif fusion approximations. It is a pleasure to thank Professors Richard C. Lewontin and Arnold J.F. Siegert for their interest and support, and Mrs. Angell Pasley for her excellent and careful typing. I . PRELIMINARIES 1. Terminology and Examples Consider an experiment specified by: a) the experiment's outcomes, ~, forming the space S; b) certain subsets of S (called events) and by the probabilities of these events.
Del 15 - Lecture Notes in Biomathematics
Selection in One- and Two-Locus Systems
Häftad, Engelska, 1977
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Most of these notes were presented as part of a two-quarter course on theoretical population genetics at The University of Chicago. Almost all the students were either undergraduates in mathematics or graduate students in the biological sciences. The only prerequisites were calculus and matrices. As is done in these notes, biological background and additional mathematical techniques were covered when they were required. I have included the relevant problems assigned in the course. My aim in these notes is to formulate the various models fairly generally, making the biological assumptions quite explicit, and to perform the analyses relatively rigorously. I hope the choice and treatment of topics will enable the reader to understand and evaluate detailed analyses of specific models and applications in the literature. No attempt has been made to review the literature or to assign credit. Most of the references are to papers directly germane to the subjects and approaches covered here. Frequency of reference is not intended to reflect proportionate contribution. I am very grateful to Professor James F. Crow for helpful comments and to Mrs. Adelaide Jaffe for her excellent typing. I thank the National Science Foundation for its support (Grant No. DEB76-01550).
Del 16 - Lecture Notes in Biomathematics
Stochastic Models for Spike Trains of Single Neurons
Häftad, Engelska, 1977
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1 Some basic neurophysiology 4 The neuron 1. 1 4 1. 1. 1 The axon 7 1. 1. 2 The synapse 9 12 1. 1. 3 The soma 1. 1. 4 The dendrites 13 13 1. 2 Types of neurons 2 Signals in the nervous system 14 2. 1 Action potentials as point events - point processes in the nervous system 15 18 2. 2 Spontaneous activi~ in neurons 3 Stochastic modelling of single neuron spike trains 19 3. 1 Characteristics of a neuron spike train 19 3. 2 The mathematical neuron 23 4 Superposition models 26 4. 1 superposition of renewal processes 26 4. 2 Superposition of stationary point processe- limiting behaviour 34 4. 2. 1 Palm functions 35 4. 2. 2 Asymptotic behaviour of n stationary point processes superposed 36 4. 3 Superposition models of neuron spike trains 37 4. 3. 1 Model 4. 1 39 4. 3. 2 Model 4. 2 - A superposition model with 40 two input channels 40 4. 3. 3 Model 4. 3 4. 4 Discussion 41 43 5 Deletion models 5. 1 Deletion models with 1nd~endent interaction of excitatory and inhibitory sequences 44 VI 5. 1. 1 Model 5. 1 The basic deletion model 45 5. 1. 2 Higher-order properties of the sequence of r-events 55 5. 1. 3 Extended version of Model 5. 1 - Model 60 5. 2 5. 2 Models with dependent interaction of excitatory and inhibitory sequences - MOdels 5. 3 and 5.
Del 17 - Lecture Notes in Biomathematics
Stochastic Problems in Population Genetics
Häftad, Engelska, 1977
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These are" notes based on courses in Theoretical Population Genetics given at the University of Texas at Houston during the winter quarter, 1974, and at the University of Wisconsin during the fall semester, 1976. These notes explore problems of population genetics and evolution involving stochastic processes. Biological models and various mathematical techniques are discussed. Special emphasis is given to the diffusion method and an attempt is made to emphasize the underlying unity of various problems based on the Kolmogorov backward equation. A particular effort was made to make the subject accessible to biology students who are not familiar with stochastic processes. The references are not exhaustive but were chosen to provide a starting point for the reader interested in pursuing the subject further. Acknowledgement I would like to use this opportunity to express my thanks to Drs. J. F. Crow, M. Nei and W. J. Schull for their hospitality during my stays at their universities. I am indebted to Dr. M. Kimura for his continuous encouragement. My thanks also go to the small but resolute groups of.students, visitors and colleagues whose enthusiasm was a great source of encouragement. I am especially obliged to Dr. Martin Curie-Cohen and Dr. Crow for reading a large part eX the manuscript and making many valuable comments. Special gratitude is expressed to Miss Sumiko Imamiya for her patience and endurance and for her efficient preparation of the manuscript.
Del 18 - Lecture Notes in Biomathematics
Mathematics and the Life Sciences
Selected Lectures, Canadian Mathematical Congress, August 1975
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For two weeks in August, 1975 more than 140 mathematicians and other scientists gathered at the Universite de Sherbrooke. The occasion was the 15th Biennial Seminar of the Canadian Mathematical Congress, entitled Mathematics and the Life Sciences. Participants in this inter disciplinary gathering included researchers and graduate students in mathematics, seven different areas of biological science, physics, chemistry and medical science. Geographically, those present came from the United States and the United Kingdom as well as from academic departments and government agencies scattered across Canada. In choosing this particular interdisciplinary topic the programme committee had two chief objectives. These were to promote Canadian research in mathematical problems of the life sciences, and to encourage co-operation and exchanges between mathematical scientists" biologists and medical re searchers. To accomplish these objective the committee assembled a stim ulating programme of lectures and talks. Six principal lecturers each delivered a series of five one-hour lectures in which various aspects of the interaction between mathematics and the life sciences were considered. In addition researchers working in the areas of health, population biology, physiology and development biology and disease processes were invited to give more than 25 hours of complementary talks.
Del 19 - Lecture Notes in Biomathematics
Measuring Selection in Natural Populations
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The present volume constitutes the proceedings of the~ymposium: "Measur- ing Selection in Natural Populations", held in memory of late professor Ove Frydenberg. The Symposium took place in Sandbjerg Manor House in Southern Jutland May 10-14, 1976. The purpose of the symposium was to reflect contemporary research on the mechanisms of biological evolu- tion. The arrangement of the symposium was possible only with the assis- tance and work of many other people. Professor Ole Barndorff-Nielsen took an active part in the planning and arrangement of the symposium. Our gratitude is also due to Ellen Christensen, Jens Ole Frier, Arno Jensen, Preben Jensen and Anne Nissen, all of whom took care of the practical arrangements at the symposium. The publication was prepared with the assistance of Arno Jensen, Preben Jensen, Verner Blak Nielsen and Kirsten Svendsen. The Symposium was financed by grants from The Danish Natural Science Research Council, The Danish Atomic Energy Commission and The University of Aarhus. Aarhus, June 1977 Freddy B. Christiansen Tom M. Fenchel LIST OF PARTICIPANTS AND CONTRIBUTORS ALLARD, R.W.:Department of Genetics, University of California, Davis, California 95616, USA. ANDERSEN, A. HOLST: Department of Theoretical Statistics, Institute of Mathematics, University of Aarhus, Ny Munkegade, DK-SOOO Aarhus C, Denmark. AYALA, F.J.: Department of Genetics, University of California, Davis, California 95616, USA. BARKER, J.S.F.: Department of Animal Husbandry, University of Sydney, Sydney 2006, New South Wales, Australia.
Del 20 - Lecture Notes in Biomathematics
Integrodifferential Equations and Delay Models in Population Dynamics
Häftad, Engelska, 1977
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These notes are, for the most part, the result of a course I taught at the University of Arizona during the Spring of 1977. Their main purpose is to inves tigate the effect that delays (of Volterra integral type) have when placed in the differential models of mathematical ecology, as far as stability of equilibria and the nature of oscillations of species densities are concerned. A secondary pur pose of the course out of which they evolved was to give students an (at least elementary) introduction to some mathematical modeling in ecology as well as to some purely mathematical subjects, such as stability theory for integrodifferentia1 systems, bifurcation theory, and some simple topics in perturbation theory. The choice of topics of course reflects my personal interests; and while these notes were not meant to exhaust the topics covered, I think they and the list of refer ences come close to covering the literature to date, as far as integrodifferentia1 models in ecology are concerned. I would like to thank the students who took the course and consequently gave me the opportunity and stimulus to organize these notes. Special thanks go to Professor Paul Fife and Dr. George Swan who also sat in the course and were quite helpful with their comments and observations. Also deserving thanks are Professor Robert O'Malley and Ms. Louise C. Fields of the Applied Mathematics Program here at the University of Arizona. Ms. Fields did an outstandingly efficient and accu rate typing of the manuscript.
Del 21 - Lecture Notes in Biomathematics
Theoretical Approaches to Complex Systems
Proceedings, Tübingen, June 11–12, 1977
Häftad, Engelska, 1978
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This volume is the issue of the symposium 'Theoretical Approaches to Complex systems', held in Tlibingen on June 11 and 12, 1977. The meeting was dedicated to the late Ernst Pfaffelhuber of the Institute for Information Sciences, University of Tlibingen, in memory of his con- tributions to information theory, physics, and biology and of his many fruitful contacts with the Tlibingen scientists gathered on this occa- sion. We would like to express our gratitude to all participants for their willingness to offer a weekend, for their help and the care they took in writing a contribution to this volume. R. Heim G. Palm CONTENTS Introduction H. Hahn 3 Existence Criteria for Bifurcations as Stability Criteria for Critical Nonlinear Control Systems G. Palm Asymptotic System Analysis and Goal Functions 51 O. E. ROssler and P. J. Ortoleva 57 Strange Attractors in 3-Variable Reaction Systems w. u. an der Heiden 75 Structures of Excitation and Inhibition T. Poggio and V. Torre 89 A new Approach to Synaptic Interactions W. Reichardt Figure-Ground Discrimination by the Visual **...**. *. ** 117 System of the Fly M. Conrad Evolution of the Adaptive Landscape ********. *. **** 147 V.Braitenberg Cell Assemblies in the Cerebral Cortex *. ****. *. ***** 171 D. Varju Excitatory and Inhibitory Processes Giving Rise ********* 189 to the Delayed Response in the Retinal Ganglion Cell of the Frog R. Heim 209 On Constrained Transinformation Functions M.
Del 22 - Lecture Notes in Biomathematics
Golden Age of Theoretical Ecology: 1923–1940
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This is the part of any book where the authors usually discuss why they wrote it. We hope, however, that the text will justifY itself. In fact, any well-trained ecologist will immediately grasp the significance of these seminal works. We have therefore tried to keep our interpretive comments to a minimum. Students of "modern" theoretical ecology will want to contrast the papers in this collection with their modern derivatives. We believe that those who do so will be surprised, if not amazed, by the ecological sophistication and intellectual power of the earlier works. They will stand as a challenge to those who study them, and we hope, provide a standard for the quality of their work. By presenting this collection of works, most of them not easily available and/or for the first time in English, we hope to help them attain the high level of recognition they deserve. We are also enabling readers not sufficiently familiar with Italian to acquire enough of a background to properly follow the works in French not presented here by including Volterra's "Variazioni e fluttuazioni del numero d' indi vidui in specie animali convi venti" (1927), still available. in the original edition.
Del 23 - Lecture Notes in Biomathematics
Geometrical Probability and Biological Structures: Buffon’s 200th Anniversary
Proceedings of the Buffon Bicentenary Symposium on Geometrical Probability, Image Analysis, Mathematical Stereology, and Their Relevance to the Determination of Biological Structures, Held in Paris, June 1977
Häftad, Engelska, 1978
550 kr
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L'etude morphologique des tissus pathologiques en microscopie d' elements: optique comporte schematiquement deux types 1) les cellules, 2) les Fibres conjonctives intercellulaires. Ces Fibres, qui constituent Ie "squelette mou", servent de support aux elements cellwlaires, contribuent a maintenir l'architecture des tissu, et interviennent dans la circulation des liquides intercellu- laires. La quantite,la structure et la repartition de ces Fibres sont etroitement liees a l'activite physiologique des populations cellulaires et a leur pathologie.L'etude de leurs modiFications est tres utile dans Ie diagnostic d'une lesion, mais la complexite et la variete des images de ces reseaux Fibrillaires rend leur description et leur classiFication tres diFFiciles. L'application des methodes morphometriques peut etre un moyen utile pour obtenir des parametres objectiFs susceptibles de preciser ces structures. C'est uniquement l'etude des Fibres conjonctives que nous envisa- gerons pour Ie moment.L'emploi,en pratique courante,des methodes morphometriques comme aide au diagnostic anatomo-pathologique,ne peut se concevoir que si les techniques utilisees sont relativement simples et surtout rapides. Dans ce but,nous avons entrepris,depuis I973,avec les moyens du Laboratoire, la realisation d'un appareil automatique specialement con~u pour essayer de repondre aux exigences particulieres de ce type d' e'tudes.
Del 24 - Lecture Notes in Biomathematics
Measurement of Biological Shape and Shape Change
Häftad, Engelska, 1978
550 kr
Skickas inom 10-15 vardagar
Del 25 - Lecture Notes in Biomathematics
Competition for Space and the Structure of Ecological Communities
Häftad, Engelska, 1978
550 kr
Skickas inom 10-15 vardagar
This volume is an investigation of interspecific competition for space, particularly among sessile organisms, both plant and animal, and its consequences for community structure. While my own contribu tion ----and the bulk of this volume --- lies in mathematical analysis of the phenomenon, I have also tried to summarize the most important natural historical aspects of these communities, and have devoted much effort to relating the mathematical results to observations of the natural world. Thus, the volume has both a synthetic and an analytic aspect. On the one hand, I have been struck by certain similarities among many communities, from forests to mussel beds, in which spatial com petition is important. On the other hand, I have analyzed this pheno menon by means of reaction-dispersal models. Finally, the mathematical analysis has suggested a conceptual framework for these communities which, I believe, further unifies and illuminates the field data. A focal perception of this work is that, just as niche relations provide an appropriate expression of the influence of resource compe tition on community structure, so do dominance relations provide an appropriate expression of the influence of spatial competition.
Del 26 - Lecture Notes in Biomathematics
Questions of Uniqueness and Resolution in Reconstruction from Projections
Häftad, Engelska, 1978
550 kr
Skickas inom 10-15 vardagar
Reconstruction from projections has revolutionized radiology and has now become one of the most important tools of medical diagnosis The E. M. I. Scanner is one example. In this text, some fundamental theoretical and practical questions are resolved. Despite recent research activity in the area, the crucial subject of the uniqueness of the reconstruction and the effect of noise in the data posed some unsettled fundamental questions. In particular, Kennan Smith proved that if we describe an object by a C^inf_o function, i.e., infinitely differentiable with compact support, then there are other objects with the same shape, i.e., support, which can differ almost arbitrarily and still have the same projections in finitely many directions. On the other hand, he proved that objects in finite dimensional function spaces are uniquely determined by a single projection for almost all angles, i.e., except on a set of measure zero. Along these lines, Herman and Rowland in "Three Methods for reconstructing objects from x-rays: a comparative study" (1973) showed that reconstructions obtained from the commonly used algorithms can grossly misrepresent the object and that the algorithm which produced the best reconstruction when using noiseless data gave unsatisfactory results with noisy data. Equally important are reports in Science, and personal communications by radiologists indicating that in medical practice failure rates of reconstruction vary from four to twenty percent. within this work, the mathematical dilemma posed by Kennan Smith's result is discussed and clarified.
Del 27 - Lecture Notes in Biomathematics
Time Lags in Biological Models
Häftad, Engelska, 1978
550 kr
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In many biological models it is necessary to allow the rates of change of the variables to depend on the past history, rather than only the current values, of the variables. The models may require discrete lags, with the use of delay-differential equations, or distributed lags, with the use of integro-differential equations. In these lecture notes I discuss the reasons for including lags, especially distributed lags, in biological models. These reasons may be inherent in the system studied, or may be the result of simplifying assumptions made in the model used. I examine some of the techniques available for studying the solution of the equations. A large proportion of the material presented relates to a special method that can be applied to a particular class of distributed lags. This method uses an extended set of ordinary differential equations. I examine the local stability of equilibrium points, and the existence and frequency of periodic solutions. I discuss the qualitative effects of lags, and how these differ according to the choice of discrete or distributed lag. The models studied are drawn from the population dynamiCS of single species (logistic growth, the chemostat) and of interacting pairs of species (predation, mutualism), from cell population dynamiCS (haemopoiesis) and from biochemical kinetics (the Goodwin oscillator). The last chapter is devoted to a population model employing difference equations. All these models include non-linear terms.
Del 28 - Lecture Notes in Biomathematics
Mathematical Aspects of Reacting and Diffusing Systems
Häftad, Engelska, 1979
550 kr
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Modeling and analyzing the dynamics of chemical mixtures by means of differ- tial equations is one of the prime concerns of chemical engineering theorists. These equations often take the form of systems of nonlinear parabolic partial d- ferential equations, or reaction-diffusion equations, when there is diffusion of chemical substances involved. A good overview of this endeavor can be had by re- ing the two volumes by R. Aris (1975), who himself was one of the main contributors to the theory. Enthusiasm for the models developed has been shared by parts of the mathematical community, and these models have, in fact, provided motivation for some beautiful mathematical results. There are analogies between chemical reactors and certain biological systems. One such analogy is rather obvious: a single living organism is a dynamic structure built of molecules and ions, many of which react and diffuse. Other analogies are less obvious; for example, the electric potential of a membrane can diffuse like a chemical, and of course can interact with real chemical species (ions) which are transported through the membrane. These facts gave rise to Hodgkin's and Huxley's celebrated model for the propagation of nerve signals. On the level of populations, individuals interact and move about, and so it is not surprising that here, again, the simplest continuous space-time interaction-migration models have the same g- eral appearance as those for diffusing and reacting chemical systems.
Del 29 - Lecture Notes in Biomathematics
Kinetic Logic: A Boolean Approach to the Analysis of Complex Regulatory Systems
Proceedings of the EMBO Course “Formal Analysis of Genetic Regulation”, Held in Brussels, September 6–16, 1977
Häftad, Engelska, 1979
550 kr
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The E M B 0 course on "Formal Analysis of Genetic Regulation" A course entitled "Formal analysis of Genetic Regulation" was held at the University of Brussels from 6 to 16 September 1977 under the auspices of EMBO (European Molecular Biology Organization). As indicated by the title of the book (but not explicitly enough by the title of the course), the main emphasis was put on a dynamic analysis of systems using logical methods, that is, methods in which functions and variables take only a limited number of values - typically two. In this respect, this course was complementary to an EMBO course using continuous methods which was held some months later in Israel by Prof. Segel. People from four very different laboratories took an active part in teaching our course in Brussels : Drs Anne LEUSSLER and Philippe VAN HAM, from the Laboratory of Prof. Jean FLORINE (Laboratoire des Systemes logiques et numeriques, Faculte des Sciences appliquees, Universite Libre de Bruxelles). Dr Stuart KAUFFMAN (Dept. of Biochemistry and Biophysics, School of Medicine, Philadelphia). Prof. Gregoire NICOLlS (Service de Biophysique Theorique, Faculte des Sciences, Universite Libre de Bruxelles) and his temporary coworker Dr David RIGNEY (presently at the Center for Statistical Mechanics and Thermodynamics of the University of Texas at Austin, Texas).