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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Introduction

Frontmatter

Ecology in Theory and Application

Abstract
Fraser Darling (1967) wrote that “Ecology… was a bigger idea than the initiators grasped.” Darling and others, writing in the 1960’s, addressed the new demands placed upon ecologists as public awareness of an environmental crisis grew. The science of ecology had, until that time, been developed to satisfy purer, more abstract, objectives—a search for understanding and explanation. It was ill prepared to do more than provide anecdotes in support of the need to guard nature’s treasures and the “balance of nature.” Only in the last two decades have serious efforts been directed to developing the theoretical basis we need to manage natural systems from a sound ecological basis.
Simon A. Levin

Resource Management

Frontmatter

Bioeconomic Modeling and Resource Management

Abstract
These notes developed from a series of lectures given originally at the University of Bremen in June 1986. They provide an introduction to the modeling of biological resource exploitation. Some of the material is covered in considerably greater detail in my books (Clark 1976a, 1985), but other topics represent more recent research. A numerically oriented problem book (Conrad and Clark, 1987) is also available.
Colin W. Clark

Common Property and the Conservation of Natural Resources

Abstract
Biological resource modeling is a blend of population biology and resource economics. The choice of the word “resource” suggests that the emphasis is on use (presumably conservative and efficient use), rather than on permanent preservation.
Robert McKelvey

Information and Area-Wide Control in Agricultural Ecology

Abstract
Agricultural ecology is a topic which could take the entire period of time available for the Course in Mathematical Ecology offered at the ICTP. For that reason, one must carefully select topics in the lectures. The topics chosen for these lectures are motivated by questions concerning agricultural productivity in developing countries. Productivity is often hampered by pest insects, which may cause enormous crop losses during outbreaks. There is considerable need for predicting where and when outbreaks are likely to be severe and to be able to implement management strategies that are effective but not excessively costly. These sentiments are echoed in the United Nations Africa Relief Program, as reported in the New York Times on 2 June 1986. The UN General Assembly adopted an agreement on African recovery that included the following points concerned with agricultural development:
The immediate objective will be to cope with future emergencies and catastrophes through the following measures:
  • To create and sustain national emergency preparedness;
  • To institute effective early warning systems;
  • To establish flexible and efficient regional networks of crop protection
Marc Mangel

Epidemiology

Frontmatter

Three Basic Epidemiological Models

Abstract
There are three basic types of deterministic models for infectious diseases which are spread by direct person-to-person contact in a population. Here these simplest models are formulated as initial value problems for systems of ordinary differential equations and are analysed mathematically. Theorems are stated regarding the asymptotic stability regions for the equilibrium points and phase plane portraits of solution paths are presented. Parameters are estimated for various diseases and are used to compare the vaccination levels necessary for herd immunity for these diseases. Although the three models presented are simple and their mathematical analyses are elementary, these models provide notation, concepts, intuition and foundation for considering more refined models. Some possible refinements are disease-related factors such as the infectious agent, mode of transmission, latent period, infectious period, susceptibility and resistance, but also social, cultural, Ecology by providing a sound intuitive understanding and complete proofs for the three most basic epidemiological models for microparasitic infections.
Herbert W. Hethcote

The Population Biology of Parasitic Helminths in Animal Populations

Abstract
Most of the other chapters on infectious diseases in this book have concentrated on microparasites, the viruses, bacteria and protozoa, in human populations. This chapter will concentrate on mathematical models for macroparasites, the parasitic helminths, in wild, domestic and laboratory populations of animals. Instead of describing any group of models in specific detail, the development of the models from a common source is illustrated. Similarly, no formal proofs of the models’ properties are given, emphasis is instead placed upon the applications of the models to the control of diseases caused by parasitic helminths. Citations to the relevant papers form an introduction to the reader wishing to either examine the mathematical properties of the models or the more specific details of their application to a specific epidemiological problem.
A. P. Dobson

Simple Versus Complex Epidemiological Models

Abstract
Mathematical models of the population dynamics of disease can contribute to a better understanding of epidemiological patterns and disease control. Simple models with few assumptions lead to general conclusions of a qualitative nature. More detailed models are useful for quantitative conclusions. However, the comparative merits of simple versus complex epidemiological models are not always readily apparent. The problem of determining an appropriate degree of complexity is discussed in reference to age structure in a model of transmission which has been applied to the study of measles immunization.
Joan L. Aron

Periodicity in Epidemiological Models

Abstract
Various epidemiological mechanisms have been shown to lead to periodic solutions. The most direct way in which periodicity arises is through extrinsic forcing by a parameter such as the contact rate, but periodicity can also arise autonomously. Cyclic models of SIRS or SEIRS type can have periodic solutions if there is a large time delay in the removed class. Epidemiological models with nonlinear incidence of certain general forms can have periodic solutions. Some models with variable population size and disease-related deaths have periodic solutions; most of these are host-parasite models where the parasite lifetime is much shorter than that of the host. Recently, periodic solutions have been found numerically in age structured models with cross immunity between two viral strains.
Herbert W. Hethcote, Simon A. Levin

Rubella

Abstract
George Maton in 1814 realized that there was a mild illness characterized by rash, adenopathy and no fever that was distinct from scarlatina. This new disease was named rubella by Henry Veale in 1866. In 1942 an Australian ophthalmologist, Norman Gregg, noticed that German measles (rubella) infection in the first trimester of pregnancy caused serious birth defects in the offspring. The rubella virus was isolated in tissue culture in 1962 at two different laboratories. After the severe rubella epidemic in 1964, it was recognized that congenital rubella syndrome (CRS) included not only cardiovascular lesions, cataracts, deafness, mental retardation, central nervous system abnormalities and generalized growth retardation, but also bone lesions, hepatitis, meningoencephalitis, progressive rubella panencephalitis and eventual diabetes mellitus. In 1969, attenuated rubella vaccines became available for use in the United States (Cooper, 1985).
Herbert W. Hethcote

Influenza and Some Related Mathematical Models

Abstract
Despite advances in biology and medical science that have controlled many severe infectious diseases, influenza remains a recurrent problem, initiating new global pandemics because of its ability to change its form. In 1918–1919, an influenza pandemic (Spanish flu) killed about 20 million people and infected perhaps 2 billion. The special feature of this pandemic was a tendency towards bronchopneumonic complications fatal to previously healthy young adults. In Philadelphia, people were dying so quickly that bodies were stacked by the hundreds in temporary morgues, awaiting burial. Such horrible mortality caused tremendous social and economic disruption, and stimulated intensive research into the cause of the disease (Beveridge, 1977).
Wei-min Liu, Simon A. Levin

Review of Recent Models of HIV/AIDS Transmission

Abstract
HIV, the human immunodeficiency acquired immunodeficiency syndrome virus, is the etiological agent for AIDS (acquired immuno deficiency syndrome). In 1982 Gallo suggested that the cause of AIDS was likely to be a new human retrovirus and, in 1983, researchers at the Pasteur Institute under the direction of Montagnier were able to isolate a new retrovirus from a New York AIDS victim (see Barre-Sinoussi et al., 1983). In 1984, Gallo and his colleagues isolated the same type of retrovirus and proved it to be the etiological agent of AIDS (for more details see Gallo, 1986, 1987; Wong-Staal and Gallo, 1985). This virus has been estimated to kill at least 30% of those infected. By April 1988, about 58,000 individuals have died of AIDS in the United States, and the Coolfont Report (1986) predicts that by 1991 the lower bound for the cumulative number of AIDS cases will be 290,000 individuals in the United States alone. One of the biggest problems associated with HIV is that most infected individuals appear to be asymptomatic and infectious for long periods of time, with an average infectious period of at least 8 years. Furthermore, there is growing evidence that the infectiousness of individuals varies with time since infection; the amount of free virus is relatively high just after infection (Francis et al., 1984; Salahuddin et al., 1984), remains low for several years, and climbs again within a year or so of the onset of AIDS (Lange et al., 1986).
Carlos Castillo-Chavez

The Transmission Dynamics of Human Immunodeficiency Virus (HIV)

Abstract
The paper first reviews data on HIV infections and AIDS disease among homosexual men, heterosexuals, IV-drug abusers and children born to infected mothers, in both developed and devloping countries. We survey such information as is currently available about the distribution of incubation times that elapse between HIV infection and the appearance of AIDS, about the fraction of those infected with HIV who eventually go on to develop AIDS, about time-dependent patterns of infectiousness, and about distributions in rates of acquiring new sexual or needle-sharing partners.
Using this information, models for the transmission dynamics of HIV are developed, beginning with deliberately oversimplified models and progressing—on the basis of the understanding thus gained-to more complex ones. Where possible, estimates of the model’s parameters are derived from the epidemiological data, and predictions are compared with observed trends. We also combine these epidemiological models with demographic considerations, to assess the effects that heterosexually-transmitted HIV/AIDS may eventually have on rates of population growth, on age profiles, and on associated economic and social indicators, in African and other countries. The degree to which sexual or other habits must change to bring the “basic reproductive rate”, R0, of HIV infections below unity, is discussed. We conclude by outlining some research needs, both in the refinement and development of models, and in the collection of epidemiological data.
Robert M. May, Roy M. Anderson

Ecotoxicology

Frontmatter

Models in Ecotoxicology: Methodological Aspects

Abstract
The science of ecotoxicology is based to a large extent on extrapolation — extrapolation from one system and one stress to another, extrapolation from laboratory tests and microcosm studies to field situations, extrapolation across scales. Such extrapolation must be based on some underlying model or models; thus, models are an essential and ineluctable component of ecological risk assessment.
Simon A. Levin

Deterministic and Statistical Models of Chemical Fate in Aquatic Systems

Abstract
This paper has several purposes: (a) to summarize the basic models of the steady state transport and fate of chemicals in aquatic systems including uptake and distribution in the aquatic food chain, (b) to illustrate the deterministic time variable behavior of chemical fate models with several applications to the Great Lakes and (c) to develop some statistical models of chemical variability in aquatic organisms, specifically, the fish.
Robert V. Thomann

Effects of Toxicants on Aquatic Populations

Abstract
In the United States, the Environmental Protection Agency regulates chemicals under several legislative acts, two of which are the Toxic Substances Control Act (TSCA) and the Federal Insecticides, Fungicides, and Rodenticides Act (FIFRA). The Agency regulates chemicals under these acts using, in part, the assessment of risks both to humans and to the environment. A scientifically based methodology of high utility to assist in evaluating environmental risk posed by the introduction of chemicals is currently under development. The purpose of this article is to provide indications of past developments, of current theoretical research, and of directions of environmental risk assessment of chemical stress on populations. Fate and effects at the community and ecosystem level are, at this stage, only speculative.
Thomas G. Hallam, Ray R. Lassiter, S. A. L. M. Kooijman

Demography and Population Biology

Frontmatter

Mathematical Models in Plant Biology: An Overview

Abstract
The study of plants could be undertaken at essentially every level of organization within biology, from that within a cell to the entire biosphere. To cover even the main theoretical questions on these diverse levels and the mathematical approaches used to analyse them would require several volumes. My objective here is to consider a small subset of the work that has been done, dealing only with the levels normally taken as being part of the purview of ecology and touching somewhat on a few more applied problems in agriculture. I will not discuss statistical analyses of plant community assemblages nor most aspects of plant-animal interactions, such as pollination biology (Real, 1983). Biophysical approaches were reviewed earlier (Gross, 1986b). Background references should be consulted for further details (e. g. France and Thornley, 1984; Givnish, 1986a; Gross and Miura, 1986; Jean, 1984; Rose and Charles-Edwards, 1981).
Louis J. Gross

Stable Population Theory and Applications

Abstract
Demography is the study of population, primarily human population, in terms of its growth and decay, its fertility and mortality, its relative mobility, and its composition, density and size. The impact on economic, political and social components of society has always been of interest to demographers. Early studies of demography as a science can be traced back to the Roman Empire, where in the year 225 A.D., the mortality schedules of Macer and Ulpian were the first published documents on the subject.
John Impagliazzo

Stage Structure Models Applied in Evolutionary Ecology

Abstract
Mathematical models of physiologically structured populations are now well established within the mainstream of theoretical ecology (Metz and Diekmann, 1986, and references therein), but to date their utilisation in many areas of ecology has been restricted by two types of difficulty. First, the numerical solution of the partial differential equations that arise naturally in the description of many structured populations is far from straightforward, and although promising methods are currently being developed (e.g. de Roos, 1988), the numerically unsophisticated worker does not have ready access to well-tested “off-the-shelf” computer packages such as are available for models posed in terms of ordinary differential equations, difference equations, or Leslie matrices. Second, practical applications of structured models demand large quantities of biological information, and it is seldom easy to formulate models that only require parameters which can be calculated from existing data.
R. M. Nisbet, W. S. C. Gurney, J. A. J. Metz

Some Applications of Structured Models in Population Dynamics

Abstract
It is now a well-known fact that age or size structure often affects qualitative changes in the dynamics of population models (see Nisbet and Gurney in Mathematical Ecology. An Introduction, eds. T.G. Hallam and S.A. Levin 1986). However the incorporation of age or size structure leads to infinite dimensional dynamical systems that are difficult to analyze. Furthermore, in some cases increased detail may reduce predictive capability because of problem of parameter estimation and error propagation. Because of this difficulty, what I call the “science of biological aggregation” has responded with systematic attempts to develop models of reduced mathematical complexity that do not sacrifice biological realism. In many instances, a minimal level of detail is required; further aggregation results in the loss of vital information and may lead to erroneous conclusions. Successful simplified realistic models have to be less aggregated.
Carlos Castillo-Chavez

Backmatter

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