Introduction to Reaction-Diffusion Equations
Introduction to Reaction-Diffusion Equations

Author: King-Yeung Lam

Publisher: Springer Nature

Published: 2022-12-01

Total Pages: 316

ISBN-13: 3031204220

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This book introduces some basic mathematical tools in reaction-diffusion models, with applications to spatial ecology and evolutionary biology. It is divided into four parts. The first part is an introduction to the maximum principle, the theory of principal eigenvalues for elliptic and periodic-parabolic equations and systems, and the theory of principal Floquet bundles. The second part concerns the applications in spatial ecology. We discuss the dynamics of a single species and two competing species, as well as some recent progress on N competing species in bounded domains. Some related results on stream populations and phytoplankton populations are also included. We also discuss the spreading properties of a single species in an unbounded spatial domain, as modeled by the Fisher-KPP equation. The third part concerns the applications in evolutionary biology. We describe the basic notions of adaptive dynamics, such as evolutionarily stable strategies and evolutionary branching points, in the context of a competition model of stream populations. We also discuss a class of selection-mutation models describing a population structured along a continuous phenotypical trait. The fourth part consists of several appendices, which present a self-contained treatment of some basic abstract theories in functional analysis and dynamical systems. Topics include the Krein-Rutman theorem for linear and nonlinear operators, as well as some elements of monotone dynamical systems and abstract competition systems. Most of the book is self-contained and it is aimed at graduate students and researchers who are interested in the theory and applications of reaction-diffusion equations.

The Role of Advection in a Two-Species Competition Model: A Bifurcation Approach
The Role of Advection in a Two-Species Competition Model: A Bifurcation Approach

Author: Isabel Averill

Publisher: American Mathematical Soc.

Published: 2017-01-18

Total Pages: 118

ISBN-13: 1470422026

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The effects of weak and strong advection on the dynamics of reaction-diffusion models have long been studied. In contrast, the role of intermediate advection remains poorly understood. For example, concentration phenomena can occur when advection is strong, providing a mechanism for the coexistence of multiple populations, in contrast with the situation of weak advection where coexistence may not be possible. The transition of the dynamics from weak to strong advection is generally difficult to determine. In this work the authors consider a mathematical model of two competing populations in a spatially varying but temporally constant environment, where both species have the same population dynamics but different dispersal strategies: one species adopts random dispersal, while the dispersal strategy for the other species is a combination of random dispersal and advection upward along the resource gradient. For any given diffusion rates the authors consider the bifurcation diagram of positive steady states by using the advection rate as the bifurcation parameter. This approach enables the authors to capture the change of dynamics from weak advection to strong advection. The authors determine three different types of bifurcation diagrams, depending on the difference of diffusion rates. Some exact multiplicity results about bifurcation points are also presented. The authors' results can unify some previous work and, as a case study about the role of advection, also contribute to the understanding of intermediate (relative to diffusion) advection in reaction-diffusion models.

Sugarcane
Sugarcane

Author: Paul H. Moore

Publisher: John Wiley & Sons

Published: 2013-12-06

Total Pages: 1063

ISBN-13: 1118771389

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Physiology of Sugarcane looks at the development of a suite of well-established and developing biofuels derived from sugarcane and cane-based co-products, such as bagasse. Chapters provide broad-ranging coverage of sugarcane biology, biotechnological advances, and breakthroughs in production and processing techniques. This single volume resource brings together essential information to researchers and industry personnel interested in utilizing and developing new fuels and bioproducts derived from cane crops.

Stochastic Modelling of Reaction–Diffusion Processes
Stochastic Modelling of Reaction–Diffusion Processes

Author: Radek Erban

Publisher: Cambridge University Press

Published: 2020-01-30

Total Pages: 322

ISBN-13: 1108572995

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This practical introduction to stochastic reaction-diffusion modelling is based on courses taught at the University of Oxford. The authors discuss the essence of mathematical methods which appear (under different names) in a number of interdisciplinary scientific fields bridging mathematics and computations with biology and chemistry. The book can be used both for self-study and as a supporting text for advanced undergraduate or beginning graduate-level courses in applied mathematics. New mathematical approaches are explained using simple examples of biological models, which range in size from simulations of small biomolecules to groups of animals. The book starts with stochastic modelling of chemical reactions, introducing stochastic simulation algorithms and mathematical methods for analysis of stochastic models. Different stochastic spatio-temporal models are then studied, including models of diffusion and stochastic reaction-diffusion modelling. The methods covered include molecular dynamics, Brownian dynamics, velocity jump processes and compartment-based (lattice-based) models.

The Mathematics of Diffusion
The Mathematics of Diffusion

Author: John Crank

Publisher: Oxford University Press

Published: 1979

Total Pages: 428

ISBN-13: 9780198534112

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Though it incorporates much new material, this new edition preserves the general character of the book in providing a collection of solutions of the equations of diffusion and describing how these solutions may be obtained.

Exactly Solvable Models of Biological Invasion
Exactly Solvable Models of Biological Invasion

Author: Sergei V. Petrovskii

Publisher: CRC Press

Published: 2005-07-28

Total Pages: 232

ISBN-13: 1420034960

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Much of our current knowledge on biological invasion was derived from field studies, but many recent advances relied heavily on mathematics and computing, particularly mathematical modeling. While numerical simulations are clearly a useful approach, they have some serious drawbacks. Approximations errors and the number of parameter values can have

Diffusion and Ecological Problems: Modern Perspectives
Diffusion and Ecological Problems: Modern Perspectives

Author: Akira Okubo

Publisher: Springer Science & Business Media

Published: 2013-04-17

Total Pages: 488

ISBN-13: 1475749783

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Surveying a wide variety of mathematical models of diffusion in the ecological context, this book is written with the primary intent of providing scientists, particularly physicists but also biologists, with some background of the mathematics and physics of diffusion and how they can be applied to ecological problems. Equally, this is a specialized text book for graduates interested in mathematical ecology -- assuming no more than a basic knowledge of probability and differential equations. Each chapter in this new edition has been substantially updated by appopriate leading researchers in the field and contains much new material covering recent developments.

Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water
Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water

Author: National Research Council

Publisher: National Academies Press

Published: 2011-08-27

Total Pages: 156

ISBN-13: 0309215625

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The human-mediated introduction of species to regions of the world they could never reach by natural means has had great impacts on the environment, the economy, and society. In the ocean, these invasions have long been mediated by the uptake and subsequent release of ballast water in ocean-going vessels. Increasing world trade and a concomitantly growing global shipping fleet composed of larger and faster vessels, combined with a series of prominent ballast-mediated invasions over the past two decades, have prompted active national and international interest in ballast water management. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water informs the regulation of ballast water by helping the Environnmental Protection Agency (EPA) and the U.S. Coast Guard (USCG) better understand the relationship between the concentration of living organisms in ballast water discharges and the probability of nonindigenous organisms successfully establishing populations in U.S. waters. The report evaluates the risk-release relationship in the context of differing environmental and ecological conditions,including estuarine and freshwater systems as well as the waters of the three-mile territorial sea. It recommends how various approaches can be used by regulatory agencies to best inform risk management decisions on the allowable concentrations of living organisms in discharged ballast water in order to safeguard against the establishment of new aquatic nonindigenous species, and to protect and preserve existing indigenous populations of fish, shellfish, and wildlife and other beneficial uses of the nation's waters. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water provides valuable information that can be used by federal agencies, such as the EPA, policy makers, environmental scientists, and researchers.