Search Results

Developing predictive models

E. J. Milner-Gulland and Marcus Rowcliffe

in Conservation and Sustainable Use: A Handbook of Techniques

The effective management of natural resources use requires a mechanistic understanding of the system, not just correlations between variables of the kind discussed in Chapter 4. Understanding may ... More

The effective management of natural resources use requires a mechanistic understanding of the system, not just correlations between variables of the kind discussed in Chapter 4. Understanding may simply be in the form of a conceptual model, but is much more powerful when formalized as a mathematical model. This chapter introduces methods for building a model of the system that can be used to predict future sustainability with or without management interventions. The emphasis is on the simulation of biological and bioeconomic dynamics, for which step-by-step worked examples are given. These examples start with conceptual models, then show how to formalize these as mathematical equations, build these into computer code; test model sensitivity, validity, and alternative structures; and finally, explore future scenarios. Methods for modelling stochasticity and human behaviour are also introduced, as well as the use of Bayesian methods for understanding dynamic systems and exploring management interventions.Less

Population dynamics: modelling demographic Allee effects

Franck Courchamp, Luděk Berec, and Joanna Gascoigne

in Allee Effects in Ecology and Conservation

Much of what is known about Allee effects comes from mathematical models. This chapter deals with models of demographic Allee effects. It shows what types of model are available, what their ... More

Much of what is known about Allee effects comes from mathematical models. This chapter deals with models of demographic Allee effects. It shows what types of model are available, what their underlying assumptions are, how they can be used, and how they have contributed to our understanding of the dynamical consequences of Allee effects, both in single-species populations and multiple-species communities. The chapter does not provide detailed information on how to develop and analyse any single model, nor discusses the structure and predictions of every model that has ever been used to study Allee effects. Rather, it is conceived as a sort of summary of a large amount of literature that has been published on models which address Allee effects: providing a comprehensive (if a little dense) overview of the topic.Less

Transport of Radionuclides in Groundwater

Richard B. Codell and James O. Duguid

in Radiological Risk Assessment and Environmental Analysis

Humans can be exposed to radioactive materials released to the ground by processes such as mining and milling of uranium, disposal of radioactive waste in repositories and landfills, or through ... More

Humans can be exposed to radioactive materials released to the ground by processes such as mining and milling of uranium, disposal of radioactive waste in repositories and landfills, or through accidents. This chapter covers practical problems of release of radioactive materials from sources, current practices for the characterization of groundwater transport pathways through the unsaturated and saturated zones, and the ultimate consumption of transported radionuclides by humans. It discusses the underlying principles of groundwater flow, the transport of radioactive materials dissolved or suspended in water, and the interaction and retardation of radionuclides with soil and rock along transport pathways. Finally, the chapter discusses the formulation of mathematical models for these processes and presents simplified analytical techniques for preliminary calculations of groundwater flow, contaminant transport, and doses.Less

CONSTRUCTIVE SET THEORIES AND THEIR CATEGORY-THEORETIC MODELS

Alex Simpson

in From Sets and Types to Topology and Analysis: Towards practicable foundations for constructive mathematics

This chapter advocates a pragmatic approach to constructive set theory, using axioms based solely on set-theoretic principles that are directly relevant to (constructive) mathematical practice. The ... More

This chapter advocates a pragmatic approach to constructive set theory, using axioms based solely on set-theoretic principles that are directly relevant to (constructive) mathematical practice. The aim is to leave the notion of set as unconstrained as possible, while remaining consistent with the ways in which sets are actually used in mathematical practice. Following this approach, the chapter presents theories ranging in power from weaker predicative theories to stronger impredicative ones. The theories considered all have sound and complete classes of category-theoretic models, obtained by axiomatizing the structure of an ambient category of classes together with its subcategory of sets. In certain special cases, the categories of sets have independent characterizations in familiar category-theoretic terms, and one thereby obtains a rich source of naturally occurring mathematical models for (both predicative and impredicative) constructive set theories.Less

Formal Models in Narrative Analysis

David Herman

Apostolos Doxiadis and Barry Mazur (eds)

in Circles Disturbed: The Interplay of Mathematics and Narrative

This chapter considers formal models of narrative and the nature of the theory of narrative. After discussing the diachronic and synchronic approaches to investigating the role of formal models in ... More

This chapter considers formal models of narrative and the nature of the theory of narrative. After discussing the diachronic and synchronic approaches to investigating the role of formal models in narrative analysis, the chapter looks at those ideas about models and modeling as a kind of bridge between humanistic and technoscientific discourse. It then evaluates descriptive and functional classifications of models, along with a range of perspectives on mathematical models and modeling. It also presents a case study in metanarratology, with a particular focus on modeling practices that have been brought to bear on focalization. It also analyzes some instances of the confluence of the formal study of narrative and mathematics, including the use of permutation groups, as well as the synergy between mathematically based theories of structural linguistics and early work on story grammars.Less

Mathematical Model

Joshua M. Epstein

in Agent_Zero: Toward Neurocognitive Foundations for Generative Social Science

This part of the book describes explicit mathematical models for the affective, cognitive, and social components of Agent_Zero. It first considers some underlying neuroscience of fear and the role of ... More

This part of the book describes explicit mathematical models for the affective, cognitive, and social components of Agent_Zero. It first considers some underlying neuroscience of fear and the role of the amygdala before turning to Rescorla–Wagner equations of conditioning. In particular, it explains how the fear circuit can be activated and how fear conditioning can occur unconsciously. It then reviews some standard nomenclature adopted by Ivan Pavlov in his study, Conditioned Reflexes: An Investigation of the Physiological Activity of the Cerebral Cortex, with emphasis on David Hume's “association of ideas,” the theory of conditioning, and the Rescorla–Wagner model. After examining “the passions,” the discussion focuses on reason, Agent_Zero's cognitive component, and the model's social component. The central case is that the agent initiates the group's behavior despite starting with the lowest disposition, with no initial emotional inclination, no evidence, the same threshold as all others, and no orders from above.Less

Some Growing Tips

C. J. Brainerd and V. F. Reyna

in The Science of False Memory

This chapter considers what the near future of the science of false memory may hold by exploring some emerging areas of experimentation. It focuses on three specific areas: mathematical models of ... More

This chapter considers what the near future of the science of false memory may hold by exploring some emerging areas of experimentation. It focuses on three specific areas: mathematical models of false memory, aging effects, and cognitive neuroscience.Less

Kelvin and Statistical Mechanics

Oliver Penrose

in Kelvin: Life, Labours and Legacy

Kelvin played a big part in the development of statistical mechanics, both for equilibrium and non-equilibrium. This chapter reviews these developments, taking a particular interest in Kelvin's own ... More

Kelvin played a big part in the development of statistical mechanics, both for equilibrium and non-equilibrium. This chapter reviews these developments, taking a particular interest in Kelvin's own contributions. Topics covered include Kelvin and thermoelectricity, gas modeled as a collection of molecules, the reversibility paradox, mathematical probability models, and Boltzmann's equations.Less

Mathematics and Narrative: A Narratological Perspective

Uri Margolin

Apostolos Doxiadis and Barry Mazur (eds)

in Circles Disturbed: The Interplay of Mathematics and Narrative

This chapter examines the manifold relations between narrative and mathematics from the point of view of narratology. In particular, it considers some of the ways in which we can speak of mathematics ... More

This chapter examines the manifold relations between narrative and mathematics from the point of view of narratology. In particular, it considers some of the ways in which we can speak of mathematics in literature; for example, the portrayal in a literary narrative of the destiny of an actual or fictional mathematician as a function of his intellectual endeavors; or the use of mathematical notions (infinity, infinite regression, branching time) as the key thematic element or basic initial situation of a narrative. The chapter also discusses the structural similarities and differences between how mathematical texts and narratives treat the creation of imaginary worlds, as well as the criteria of truth, levels and hierarchies of representation involved in this process. Finally, it explores the logical requirements for computer simulations of future scenarios, along with the mathematical concepts, models, and methods used in theories of narrative.Less

Imaging, Picturing, and Scaling

Bas C. van Fraassen

in Scientific Representation: Paradoxes of Perspective

Resemblance comes in, not when we are answering the question — What is representation? — but rather when we address How does this or that representation represent, and how does it succeed? The ... More

Resemblance comes in, not when we are answering the question — What is representation? — but rather when we address How does this or that representation represent, and how does it succeed? The various modes of representation here distinguished include imaging, picturing, and scaling. Imaging is representation that is effected through resemblance; picturing is imaging that carries the hallmarks of perspectivity. Mathematical representation of nature includes much imagery, and has so far always involved some features that, in retrospect, with hindsight, were seen as necessary failures in resemblance. Such representations can fall short through idealization, but can also achieve their success through systematic distortion.Less

General introduction

Baltazar D. Aguda and Avner Friedman

in Models of Cellular Regulation

This introductory chapter begins with a brief discussion of the primary goal of this book, which is to provide an account of networks of physicochemical interactions and the cellular processes that ... More

This introductory chapter begins with a brief discussion of the primary goal of this book, which is to provide an account of networks of physicochemical interactions and the cellular processes that they regulate: cell growth and division, death, differentiation, and aging. The general aim is to illustrate how mathematical models of these processes can be developed and analyzed. An overview of the chapters is presented. This is followed by a discussion on mathematical modelling of biological phenomena and a brief note on the organization and use of the book.Less

Fictions and Folk Ontology

Michael Weisberg

in Simulation and Similarity: Using Models to Understand the World

Chapter 3 develops a version of the maths view of mathematical models. On this view, mathematical and computational models are interpreted mathematical structures. However, in the last few years, a ... More

Chapter 3 develops a version of the maths view of mathematical models. On this view, mathematical and computational models are interpreted mathematical structures. However, in the last few years, a concrete or fictions view of mathematical models has become especially popular. Philosophers such as Roman Frigg, Peter Godfrey-Smith, and Arnon Levy have argued that the best way to understand mathematical models is as if they were more closely related to literary fictions than to bits of mathematics. This chapter argues against such accounts, but also shows how the maths view can be modified to take into account some of the insights of these philosophers.Less

Mathematical and computational modelling tools

Baltazar D. Aguda and Avner Friedman

in Models of Cellular Regulation

This chapter reviews chemical kinetics to illustrate the formulation of model equations for a given reaction mechanism. For spatially uniform systems, these model equations are usually ordinary ... More

This chapter reviews chemical kinetics to illustrate the formulation of model equations for a given reaction mechanism. For spatially uniform systems, these model equations are usually ordinary differential equations; but coupling of chemical reactions to physical processes such as diffusion requires the formulation of partial differential equations to describe the spatiotemporal evolution of the system. Mathematical analysis of the dynamical models involves basic concepts from ordinary and partial differential equations. Computational methods, including stochastic simulations and sources of computer software programs available free on the internet are also summarized.Less

Introduction

J. N. REDDY

in An Introduction to Nonlinear Finite Element Analysis

One of the most important thing engineers and scientists do is to model natural phenomena. They develop conceptual and mathematical models to simulate physical events, whether they are aerospace, ... More

One of the most important thing engineers and scientists do is to model natural phenomena. They develop conceptual and mathematical models to simulate physical events, whether they are aerospace, biological, chemical, geological, or mechanical. The mathematical models are developed using laws of physics and they are often described in terms of algebraic, differential, and/or integral equations relating various quantities of interest. A mathematical model of the motion of the pendulum can be constructed using the principle of conservation of linear momentum, that is, Newton's second law of motion. This chapter also discusses the traditional finite element method, nonlinear analysis, and classification of nonlinearities. The finite element method is a powerful method that can be used to perform numerical simulations of engineering problems.Less

Introduction

Michael Doebeli

in Adaptive Diversification (MPB-48)

This introductory chapter provides an overview of frequency-dependent selection—the phenomenon that the evolving population is part of the changing environment determining the evolutionary ... More

This introductory chapter provides an overview of frequency-dependent selection—the phenomenon that the evolving population is part of the changing environment determining the evolutionary trajectory. Selection is frequency-dependent if the sign and magnitude of the correlations between heritable variation and reproductive variation change as a consequence of changes in the trait distribution that are themselves generated by such correlations. From the perspective of mathematical modeling, the realm of frequency dependence in evolution is larger than the realm of situations in which selection is not frequency dependent, because the absence of frequency dependence in a mathematical model of evolution essentially means that some parameters describing certain types of biological interactions are set to zero. Thus, in a suitable parameter space, frequency independence corresponds to the region around zero, while everything else corresponds to frequency dependence. In this way, frequency-dependent selection should therefore be considered the norm, not the exception, for evolutionary processes.Less