From rainbows, river meanders, and shadows to spider webs, honeycombs, and the markings on animal coats, the visible world is full of patterns that can be described mathematically. Examining such readily observable phenomena, this book introduces readers to the beauty of nature as revealed by mathematics and the beauty of mathematics as revealed in nature. Generously illustrated, written in an informal style, and replete with examples from everyday life, Mathematics in Nature is an excellent and undaunting introduction to the ideas and methods of mathematical modeling. It illustrates how mathematics can be used to formulate and solve puzzles observed in nature and to interpret the solutions. In the process, it teaches such topics as the art of estimation and the effects of scale, particularly what happens as things get bigger. Readers will develop an understanding of the symbiosis that exists between basic scientific principles and their mathematical expressions as well as a deeper appreciation for such natural phenomena as cloud formations, halos and glories, tree heights and leaf patterns, butterfly and moth wings, and even puddles and mud cracks. Developed out of a university course, this book makes an ideal supplemental text for courses in applied mathematics and mathematical modeling. It will also appeal to mathematics educators and enthusiasts at all levels, and is designed so that it can be dipped into at leisure.
The first history of population ecology traces two generations of science and scientists from the opening of the twentieth century through 1970. Kingsland chronicles the careers of key figures and the field's theoretical, empirical, and institutional development, with special attention to tensions between the descriptive studies of field biologists and later mathematical models. This second edition includes a new afterword that brings the book up to date, with special attention to the rise of "the new natural history" and debates about ecology's future as a large-scale scientific enterprise.
A guide to using Mathematica so as to explore cellular automata within natural phenomena, such as insect colonies, bird flight paths and even DNA sequencing. Designed for physicists, life scientists, and engineers - in fact, everyone dealing with fractals - the book first introduces Mathematica before going on to provide the valuable information needed to properly motivate the code and run the simulations presented in the book. All these simulations have been tested both inside and outside the classroom setting, allowing the book's use as reference material as well as a textbook or course supplement. Packaged together with a DOS diskette enabling cross-platfform access to the code. The files will also be accessible via the World Wide Web.
This is a book about the nature of mathematical modeling, and about the kinds of techniques that are useful for modeling. The text is in four sections. The first covers exact and approximate analytical techniques; the second, numerical methods; the third, model inference based on observations; and the last, the special role of time in modeling. Each of the topics in the book would be the worthy subject of a dedicated text, but only by presenting the material in this way is it possible to make so much material accessible to so many people. Each chapter presents a concise summary of the core results in an area. The text is complemented by extensive worked problems.
This monograph offers a critical introduction to current theories of how scientific models represent their target systems. Representation is important because it allows scientists to study a model to discover features of reality. The authors provide a map of the conceptual landscape surrounding the issue of scientific representation, arguing that it consists of multiple intertwined problems. They provide an encyclopaedic overview of existing attempts to answer these questions, and they assess their strengths and weaknesses. The book also presents a comprehensive statement of their alternative proposal, the DEKI account of representation, which they have developed over the last few years. They show how the account works in the case of material as well as non-material models; how it accommodates the use of mathematics in scientific modelling; and how it sheds light on the relation between representation in science and art. The issue of representation has generated a sizeable literature, which has been growing fast in particular over the last decade. This makes it hard for novices to get a handle on the topic because so far there is no book-length introduction that would guide them through the discussion. Likewise, researchers may require a comprehensive review that they can refer to for critical evaluations. This book meets the needs of both groups.
Individual-based models are an exciting and widely used new tool for ecology. These computational models allow scientists to explore the mechanisms through which population and ecosystem ecology arises from how individuals interact with each other and their environment. This book provides the first in-depth treatment of individual-based modeling and its use to develop theoretical understanding of how ecological systems work, an approach the authors call "individual-based ecology.? Grimm and Railsback start with a general primer on modeling: how to design models that are as simple as possible while still allowing specific problems to be solved, and how to move efficiently through a cycle of pattern-oriented model design, implementation, and analysis. Next, they address the problems of theory and conceptual framework for individual-based ecology: What is "theory"? That is, how do we develop reusable models of how system dynamics arise from characteristics of individuals? What conceptual framework do we use when the classical differential equation framework no longer applies? An extensive review illustrates the ecological problems that have been addressed with individual-based models. The authors then identify how the mechanics of building and using individual-based models differ from those of traditional science, and provide guidance on formulating, programming, and analyzing models. This book will be helpful to ecologists interested in modeling, and to other scientists interested in agent-based modeling.
This book is for statistical practitioners, particularly those who design and analyze studies for survival and event history data. Building on recent developments motivated by counting process and martingale theory, it shows the reader how to extend the Cox model to analyze multiple/correlated event data using marginal and random effects. The focus is on actual data examples, the analysis and interpretation of results, and computation. The book shows how these new methods can be implemented in SAS and S-Plus, including computer code, worked examples, and data sets.
How fixed are the happenings in Nature and how are they fixed? These lectures address what our scientific successes at predicting and manipulating the world around us suggest in answer. One—very orthodox—account teaches that the sciences offer general truths that we combine with local facts to derive our expectations about what will happen, either naturally or when we build a device to design, be it a laser, a washing machine, an anti-malarial bed net, or an auction for the airwaves. In these three 2017 Carus Lectures Nancy Cartwright offers a different picture, one in which neither we, nor Nature, have such nice rules to go by. Getting real predictions about real happenings is an engineering enterprise that makes clever use of a great variety of different kinds of knowledge, with few real derivations in sight anywhere. It takes artful modeling. Orthodoxy would have it that how we do it is not reflective of how Nature does it. It is, rather, a consequence of human epistemic limitations. That, Cartwright argues, is to put our reasoning just back to front. We should read our image of what Nature is like from the way our sciences work when they work best in getting us around in it, non plump for a pre-set image of how Nature must work to derive what an ideal science, freed of human failings, would be like. Putting the order of inference right way around implies that like us, Nature too is an artful modeler. Lecture 1 is an exercise in description. It is a study of the practices of science when the sciences intersect with the world and, then, of what that world is most likely like given the successes of these practices. Millikan's famous oil drop experiment, and the range of knowledge pieced together to make it work, are used to illustrate that events in the world do not occur in patterns that can be properly described in so-called "laws of nature." Nevertheless, they yield to artful modeling. Without a huge leap of faith, that, it seems, is the most we can assume about the happenings in Nature. Lecture 2 is an exercise in metaphysics. How could the arrangements of happenings come to be that way? In answer, Cartwright urges an ontology in which powers act together in different ways depending on the arrangements they find themselves in to produce what happens. It is a metaphysics in which possibilia are real because powers and arrangement are permissive—they constrain but often do not dictate outcomes (as we see in contemporary quantum theory). Lecture 3, based on Cartwright's work on evidence-based policy and randomized controlled trials, is an exercise in the philosophy of social technology: How we can put our knowledge of powers and our skills at artful modeling to work to build more decent societies and how we can use our knowledge and skills to evaluate when our attempts are working. The lectures are important because: They offer an original view on the age-old question of scientific realism in which our knowledge is genuine, yet our scientific principles are neither true nor false but are, rather, templates for building good models. Powers are center-stage in metaphysics right now. Back-reading them from the successes of scientific practice, as Lecture 2 does, provides a new perspective on what they are and how they function. There is a loud call nowadays to make philosophy relevant to "real life." That's just what happens in Lecture 3, where Cartwright applies the lesson of Lectures 1 and 2 to argue for a serious rethink of the way that we are urged—and in some places mandated—to use evidence to predict the outcomes of our social policies.
This book develops the mathematical tools essential for students in the life sciences to describe interacting systems and predict their behavior. From predator-prey populations in an ecosystem, to hormone regulation within the body, the natural world abounds in dynamical systems that affect us profoundly. Complex feedback relations and counter-intuitive responses are common in nature; this book develops the quantitative skills needed to explore these interactions. Differential equations are the natural mathematical tool for quantifying change, and are the driving force throughout this book. The use of Euler’s method makes nonlinear examples tractable and accessible to a broad spectrum of early-stage undergraduates, thus providing a practical alternative to the procedural approach of a traditional Calculus curriculum. Tools are developed within numerous, relevant examples, with an emphasis on the construction, evaluation, and interpretation of mathematical models throughout. Encountering these concepts in context, students learn not only quantitative techniques, but how to bridge between biological and mathematical ways of thinking. Examples range broadly, exploring the dynamics of neurons and the immune system, through to population dynamics and the Google PageRank algorithm. Each scenario relies only on an interest in the natural world; no biological expertise is assumed of student or instructor. Building on a single prerequisite of Precalculus, the book suits a two-quarter sequence for first or second year undergraduates, and meets the mathematical requirements of medical school entry. The later material provides opportunities for more advanced students in both mathematics and life sciences to revisit theoretical knowledge in a rich, real-world framework. In all cases, the focus is clear: how does the math help us understand the science?
