Biomathematics – serie

This book considers many fundamental problems of molecular biology, evolution, molecular genetic organization, the structure and function of macromolecules, always with the underlying motive of developing a unified theory. It describes theoretical results as well as computational methods. The main topic of the book is to focus to develop constructive synthesis of the concepts, factual data, mathematical methods, theoretical models, scientific results, interpretations in the field of the Theory of Origin and further Evolution of Molecular Genetic Regulatory Systems, their macromolecule components and subsystems.

In 1970, Manfred Eigen initiated the study of the origin of self-reproducing systems of macromolecules and their evolution. Large-scale nucleotide sequencing (with computer methods) was introduced from 1977. The authors of this book, the first edition of which appeared (in Russian) in 1985, have been engaged in the research of the evolution of molecular genetic regulatory systems ever since those pioneering years. The book considers many fundamental problems of molecular biology, evolution, molecular genetic organization, the structure and function of macromolecules, always with the underlying motive of developing a unified theory. It describes many original, theoretical results as well as computational methods.

A basic method of analyzing particulate gene systems is the proba­ bilistic and statistical analyses. Mendel himself could not escape from an application of elementary probability analysis although he might have been unaware of this fact. Even Galtonian geneticists in the late 1800's and the early 1900's pursued problems of heredity by means of mathe­ matics and mathematical statistics. They failed to find the principles of heredity, but succeeded to establish an interdisciplinary area between mathematics and biology, which we call now Biometrics, Biometry, or Applied Statistics. A monumental work in the field of popUlation genetics was published by the late R. A. Fisher, who analyzed "the correlation among relatives" based on Mendelian gene theory (1918). This theoretical analysis over­ came "so-called blending inheritance" theory, and the orientation of Galtonian explanations for correlations among relatives for quantitative traits rapidly changed. We must not forget the experimental works of Johanson (1909) and Nilsson-Ehle (1909) which supported Mendelian gene theory. However, a large scale experiment for a test of segregation and linkage of Mendelian genes affecting quantitative traits was, prob­ ably for the first time, conducted by K. Mather and his associates and Panse in the 1940's.

The Second Autumn Course on Mathematical Ecology was held at the Intern­ ational Centre for Theoretical Physics in Trieste, Italy in November and December of 1986. During the four year period that had elapsed since the First Autumn Course on Mathematical Ecology, sufficient progress had been made in applied mathemat­ ical ecology to merit tilting the balance maintained between theoretical aspects and applications in the 1982 Course toward applications. The course format, while similar to that of the first Autumn Course on Mathematical Ecology, consequently focused upon applications of mathematical ecology. Current areas of application are almost as diverse as the spectrum covered by ecology. The topiys of this book reflect this diversity and were chosen because of perceived interest and utility to developing countries. Topical lectures began with foundational material mostly derived from Math­ ematical Ecology: An Introduction (a compilation of the lectures of the 1982 course published by Springer-Verlag in this series, Volume 17) and, when possible, progressed to the frontiers of research. In addition to the course lectures, workshops were arranged for small groups to supplement and enhance the learning experience. Other perspectives were provided through presentations by course participants and speakers at the associated Research Conference. Many of the research papers are in a companion volume, Mathematical Ecology: Proceedings Trieste 1986, published by World Scientific Press in 1988. This book is structured primarily by application area. Part II provides an introduction to mathematical and statistical applications in resource management.

This book is written by a mathematician and a theoretical biologist who have arrived at a good mutual understanding and a well worked-out common notation. The reader need hardly be convinced of the necessity of such a mutual understanding, not only for the two investigators, but also for the sciences they represent. Like Moliere's hero, geneticists are gradually beginning to understand that, unknowingly, they have been speaking in the language of cybernetics. Mathematicians are unexpec­ tedly discovering that many past and present problems and methods of genetics can be naturally formulated in the language of graph theory. In this way a powerful abstract mathematical theory suddenly finds a productive application. Moreover, in its turn, such an application be­ gins to "feed" the mathematical theory by presenting it with a number of new problems. The reader may judge for himself the fruitfulness of such mutual interaction. At the same time several important circumstances need to be men­ tioned. The formalization and rigorous formulation given here embraces not only the older problems, known by geneticists for many decades (the construction of genetic maps, the analysis of complementation, etc. ), but also comparatively new problems: the construction of partial com­ plementation maps, phylogenetic trees of proteins, etc.

Relationships between sets of variables of different kinds are of interest in many branches of science. The question of the analysis of relationships of this sort has nevertheless rather surprisingly received less attention from statisticians and others than it would seem to deserve. Of the available methods, that address­ ing the question most directly is canonical correlation analysis, here referred to for convenience as canonical analysis. Yet canonical analysis is often coolly received despite a lack of suitable alternatives. The purpose of this book is to clarify just what may and what may not be accomplished by means of canoni­ cal analysis in one field of scientific endeavor. Canonical analysis is concerned with reducing the correlation structure be­ tween two (or more) sets of variables to its simplest possible form. After a review of the nature and properties of canonical analysis, an assessment of the method as an exploratory tool of use in ecological investigations is made. Applications of canonical analysis to several sets of ecological data are described and discussed with this objective in mind. The examples are drawn largely from plant ecology. The position is adopted that canonical analysis exists primarily to be used; the examples are accordingly worked through in some detail with the aim of showing how canonical analysis can contribute towards the attainment of ecological goals, as well as to indicate the kind and extent of the insight afforded.

There isprobably no more appropriate location to hold a course on mathematical ecology than Italy, the countryofVito Volterra, a founding father ofthe subject. The Trieste 1982Autumn Course on Mathematical Ecology consisted of four weeksofvery concentrated scholasticism and aestheticism. The first weeks were devoted to fundamentals and principles ofmathematicalecology.A nucleusofthe material from the lectures presented during this period constitutes this book. The final week and a half of the Course was apportioned to the Trieste Research Conference on Mathematical Ecology whose proceedings have been published as Volume 54, Lecture Notes in Biomathematics, Springer-Verlag. The objectivesofthe first portionofthe course wereambitious and, probably, unattainable. Basic principles of the areas of physiological, population, com- munitY,and ecosystem ecology that have solid ecological and mathematical foundations were to be presented. Classical terminology was to be introduced, important fundamental topics were to be developed, some past and some current problems of interest were to be presented, and directions for possible research were to be provided.Due to time constraints, the coverage could not be encyclopedic;many areas covered already have merited treatises of book length. Consequently, preliminary foundation material was covered in some detail, but subject overviewsand area syntheseswerepresented when research frontiers were being discussed. These lecture notes reflect this course philosophy.

In May 1984 the Swedish Council for Scientific Research convened a small group of investigators at the scientific research station at Abisko, Sweden, for the purpose of examining various conceptual and mathematical views of the evolution of complex systems. The stated theme of the meeting was deliberately kept vague, with only the purpose of discussing alternative mathematically based approaches to the modeling of evolving processes being given as a guideline to the participants. In order to limit the scope to some degree, it was decided to emphasize living rather than nonliving processes and to invite participants from a range of disciplinary specialities spanning the spectrum from pure and applied mathematics to geography and analytic philosophy. The results of the meeting were quite extraordinary; while there was no intent to focus the papers and discussion into predefined channels, an immediate self-organizing effect took place and the deliberations quickly oriented themselves into three main streams: conceptual and formal structures for characterizing sys­ tem complexity; evolutionary processes in biology and ecology; the emergence of complexity through evolution in natural lan­ guages. The chapters presented in this volume are not the proceed­ ings of the meeting. Following the meeting, the organizers felt that the ideas and spirit of the gathering should be preserved in some written form, so the participants were each requested to produce a chapter, explicating the views they presented at Abisko, written specifically for this volume. The results of this exercise form the volume you hold in your hand.

Infectious diseases are transmitted through variousdifferent mechanisms including person to personinteractions, by insect vectors and via verticaltransmission from a parent to an unborn offspring. Thepopulation dynamics of such disease transmission can be verycomplicated and the development of rational strategies forcontrolling and preventing the spread of these diseasesrequires careful modeling and analysis.The book describes current methods for formulating modelsand analyzing the dynamics of the propagation of diseaseswhich include vertical transmission as one of the mechanismsfor their spread. Generic models that describe broad classesof diseases as well as models that are tailored to thedynamics of a specific infection are formulated andanalyzed. The effects of incubation periods, maturationdelays, and age-structure, interactions between diseasetransmission and demographic changes, population crowding,spatial spread, chaotic dynamic behavior, seasonalperiodicities and discrete time interval events are studiedwithin the context of specific disease transmission models.No previous background in disease transmission modeling andanalysis is assumedand the required biological concepts andmathematical methods are gradually introduced within thecontext of specific disease transmission models. Graphs arewidely used to illustrate and explain the modelingassumptions and results.REMARKS: NOTE: the authors have supplied variants on thepromotion text that are more suitable for promotionindifferent fields (by virtue of different emphasis in thecontent). They are not enclosed, but in the mathematicseditorial.

Mathematical methods have been applied successfully to population genet­ ics for a long time. Even the quite elementary ideas used initially proved amazingly effective. For example, the famous Hardy-Weinberg Law (1908) is basic to many calculations in population genetics. The mathematics in the classical works of Fisher, Haldane and Wright was also not very complicated but was of great help for the theoretical understanding of evolutionary pro­ cesses. More recently, the methods of mathematical genetics have become more sophisticated. In use are probability theory, stochastic processes, non­ linear differential and difference equations and nonassociative algebras. First contacts with topology have been established. Now in addition to the tra­ ditional movement of mathematics for genetics, inspiration is flowing in the opposite direction, yielding mathematics from genetics. The present mono­ grapll reflects to some degree both patterns but especially the latter one. A pioneer of this synthesis was S. N. Bernstein. He raised-and partially solved- -the problem of characterizing all stationary evolutionary operators, and this work was continued by the author in a series of papers (1971-1979). This problem has not been completely solved, but it appears that only cer­ tain operators devoid of any biological significance remain to be addressed. The results of these studies appear in chapters 4 and 5. The necessary alge­ braic preliminaries are described in chapter 3 after some elementary models in chapter 2.

This book covers those areas of theoretical population genetics that can be investigated rigorously by elementary mathematical methods. I have tried to formulate the various models fairly generally and to state the biological as­ sumptions quite explicitly. I hope the choice and treatment of topics will en­ able the reader to understand and evaluate detailed analyses of many specific models and applications in the literature. Models in population genetics are highly idealized, often even over­ idealized, and their connection with observation is frequently remote. Further­ more, it is not practicable to measure the parameters and variables in these models with high accuracy. These regrettable circumstances amply justify the use of appropriate, lucid, and rigorous approximations in the analysis of our models, and such approximations are often illuminating even when exact solu­ tions are available. However, our empirical and theoretical limitations justify neither opaque, incomplete formulations nor unconvincing, inadequate analy­ ses, for these may produce uninterpretable, misleading, or erroneous results. Intuition is a principal source of ideas for the construction and investigation of models, but it can replace neither clear formulation nor careful analysis. Fisher (1930; 1958, pp. x, 23-24, 38) not only espoused similar ideas, but he recognized also that our concepts of intuition and rigor must evolve in time. The book is neither a review of the literature nor a compendium of results. The material is almost entirely self-contained. The first eight chapters are a thoroughly revised and greatly extended version of my published lecture notes (Nagylaki, 1977a).

This volume is a revised and enlarged version of Chapters 1 and 2 of a book with the same title, published in Romanian in 1968. The revision resulted in a new book which has been divided into two parts because of the large amount of new material. The present part is intended to introduce mathematicians and biologists with a strong mathematical and probabilistic background to the study of stochastic processes. We hope some readers will be able to discover by themselves the new features of our treatment such as the inclusion of some unusual topics, the special attention paid to some usual topics, and the grouping of the material. We draw the reader's attention to the numbering, because there are structural differences between the two parts. In Part I there are Chapters, Sections, Subsections, Paragraphs and Subparagraphs. Thus the numbering a. b. c. d. e refers to Subparagraph e of Paragraph d of Subsection c of Section b of Chapter a. Definitions, theorems lemmas and propositions are numbered a. b. n, n = 1,2, . . . , where a indicates the chapter and b the section. In Part II there are Sections, Subsections, Paragraphs, and SUbparagraphs. Thus the numbering a. b. c. d refers to Subparagraph d of Paragraph c of Subsection b of Section a. Theorems and lemmas are numbered a. n, n = 1, 2, . . . , where a indicates the section.

This volume is a revised and enlarged version of Chapter 3 of. a book with the same title, published in Romanian in 1968. The revision resulted in a new book which has been divided into two of the large amount of new material. The whole book parts because is intended to introduce mathematicians and biologists with a strong mathematical background to the study of stochastic processes and their applications in biological sciences. It is meant to serve both as a textbook and a survey of recent developments. Biology studies complex situations and therefore needs skilful methods of abstraction. Stochastic models, being both vigorous in their specification and flexible in their manipulation, are the most suitable tools for studying such situations. This circumstance deter­ mined the writing of this volume which represents a comprehensive cross section of modern biological problems on the theory of stochastic processes. Because of the way some specific problems have been treat­ ed, this volume may also be useful to research scientists in any other field of science, interested in the possibilities and results of stochastic modelling. To understand the material presented, the reader needs to be acquainted with probability theory, as given in a sound introductory course, and be capable of abstraction.

This book is an outgrowth of one phase of an upper-division course on quantitative ecology, given each year for the past eight at Berkeley. I am most grateful to the students in that course and to many graduate students in the Berkeley Department of Zoology and Colleges of Engineering and Natural Resources whose spirited discussions inspired much of the book's content. I also am deeply grateful to those faculty colleagues with whom, at one time or another, I have shared courses or seminars in ecology or population biology, D.M. Auslander, L. Demetrius, G. Oster, O.H. Paris, F.A. Pitelka, A.M. Schultz, Y. Takahashi, D.B. Tyler, and P. Vogelhut, all of whom contributed substantially to the development of my thinking in those fields, to my Depart­ mental colleagues E. Polak and A.J. Thomasian, who guided me into the litera­ ture on numerical methods and stochastic processes, and to the graduate students who at one time or another have worked with me on population-biology projects, L.M. Brodnax, S-P. Chan, A. Elterman, G.C. Ferrell, D. Green, C. Hayashi, K-L. Lee, W.F. Martin Jr., D. May, J. Stamnes, G.E. Swanson, and I. Weeks, who, together, undoubtedly provided me with the greatest inspiration. I am indebted to the copy-editing and production staff of Springer-Verlag, especially to Ms. M. Muzeniek, for their diligence and skill, and to Mrs. Alice Peters, biomathematics editor, for her patience.

Mathematical Demography, the study of population and its analysis through mathematical models, has received increased interest in the mathematical com­ munity in recent years. It was not until the twentieth century, however, that the study of population, predominantly human population, achieved its math­ ematical character. The subject of mathematical demography can be viewed from either a deterministic viewpoint or from a stochastic viewpoint. For the sake of brevity, stochastic models are not included in this work. It is, therefore, my intention to consider only established deterministic models in this discussion, starting with the life table as the earliest model, to a generalized matrix model which is developed in this treatise. These deterministic models provide sufficient de­ velopment and conclusions to formulate sound mathematical population analy­ sis and estimates of population projections. It should be noted that although the subject of mathematical demography focuses on human populations, the development and results may be applied to any population as long as the preconditions that make the model valid are maintained. Information concerning mathematical demography is at best fragmented.

According to a recent report of the United States Census Bureau, world population as of June 30, 1983, was estimated at about 4. 7 billion people; of this total, an estimated 82 million had been added in the previous year. World population in 1950 was estimated at about 2. 5 billion; consequently, if 82 million poeple are added to the world population in each of the coming four years, population size will be double that of 1950. Another way of viewing the yearly increase in world population is to compare it to 234 million, the estimated current population of the United States. If the excess of births over deaths continues, a group of young people equivalent to the population of the United States will be added to the world population about every 2. 85 years. Although the rate of increase in world population has slowed since the midsixties, it seems likely that large numbers of infants will be added to the population each year for the foreseeable future. A large current world population together with a high likelihood of sub­ stantial increments in size every year has prompted public and scholarly recognition of population as a practical problem. Tangible evidence in the public domain that population is being increasingly viewed as a problem is provided by the fact that many governments around the world either have or plan to implement policies regarding population. Evidence of scholarly concern is provided by an increasing flow of publications dealing with population.

Cybernetics, a science concerned with understanding how systems are regulated, has reflected the preoccupations of the century in which it was born. Regulation is important in twentieth century society, where both machines and social organizations are complex. Cybernetics focused on and became primarily associated with the homeostasis or stability of system behavior and with the negative feedbacks that stabilize systems. It paid less attention to the processes opposite to negative feedback, the positive feedback processes that act to change systems. We attempt to redress the balance here by illustrating the enormous importance of positive feedbacks in natural systems. In an article in the American Scientist in 1963, Maruyama called for increased attention to this topic, noting that processes of change could occur when a "deviation in anyone component of the system caused deviations in other components that acted back on the first component to reinforce of amplify the initial deviation." The deviation amplification is the result of positive feedback among system components. Maruyama demonstrated by numerous examples that the neglect of such processes was unjustified and suggested that a new branch of cybernetics, "the second cybernetics," be devoted to their study.

Food webs hold a central place in ecology. They describe which organisms feed on which others in natural habitats. This book describes recently discovered empirical regularities in real food webs: it proposes a novel theory unifying many of these regularities, as well as extensive empirical data. After a general introduction, reviewing the empirical and theoretical discoveries about food webs, the second portion of the book shows that community food webs obey several striking phenomenological regularities. Some of these unify, regardless of habitat. Others differentiate, showing that habitat significantly influences structure. The third portion of the book presents a theoretical analysis of some of the unifying empirical regularities. The fourth portion of the book presents 113 community food webs. Collected from scattered sources and carefully edited, they are the empirical basis for the results in the volume. The largest available set of data on community food webs provides a valuable foundation for future studies of community food webs. The book is intended for graduate students, teachers and researchers primarily in ecology. The theoretical portions of the book provide materials useful to teachers of applied combinatorics, in particular, random graphs. Researchers in random graphs will find here unsolved mathematical problems.