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Exactly Solvable Models of Biological Invasion

Author: Sergei V. Petrovskii

Publisher: CRC Press

Published: 2005-07-28

Total Pages: 242

ISBN-13: 9781584885214

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 a significant impact on the simulation results, the extent of which often remains obscure. Such difficulties do not arise, however, when the problem can be solved analytically. Exactly Solvable Models of Biological Invasion demonstrates the advantages and methods of obtaining exact solutions of partial differential equations that describe nonlinear problems encountered in the study of invasive species spread. With emphasis on PDEs of diffusion-reaction type, the authors present a comprehensive collection of exactly solvable models and a unified, self-contained description of the relevant mathematical methods. In doing so, they also provide new insight into important issues such as the impact of the Allee effect, the impact of predation, and the interplay between different modes of species dispersal. Full calculation details make this presentation accessible to biologists as well as applied mathematicians, and a range of ecological examples and applications demonstrate the utility of exact methods in practice. Exact solutions provide an immediate, complete description of system dynamics for a wide class of initial conditions and serve as a convenient tool for testing numerical algorithms and codes used in more specialized studies. This book lays the groundwork for bringing the power of exactly solvable models to bear on real-world ecological problems.

The Mathematics Behind Biological Invasions

Author: Mark A. Lewis

Publisher: Springer

Published: 2016-05-05

Total Pages: 373

ISBN-13: 3319320432

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This book investigates the mathematical analysis of biological invasions. Unlike purely qualitative treatments of ecology, it draws on mathematical theory and methods, equipping the reader with sharp tools and rigorous methodology. Subjects include invasion dynamics, species interactions, population spread, long-distance dispersal, stochastic effects, risk analysis, and optimal responses to invaders. While based on the theory of dynamical systems, including partial differential equations and integrodifference equations, the book also draws on information theory, machine learning, Monte Carlo methods, optimal control, statistics, and stochastic processes. Applications to real biological invasions are included throughout. Ultimately, the book imparts a powerful principle: that by bringing ecology and mathematics together, researchers can uncover new understanding of, and effective response strategies to, biological invasions. It is suitable for graduate students and established researchers in mathematical ecology.

Approaches to Mathematical Modeling of Biological Invasions

Author: Merlin C. Köhnke

Publisher:

Published: 2020-08-21

Total Pages: 246

ISBN-13: 9783832551513

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With potentially dangerous biological invasions increasing due to rises in international trade and travel, research in invasion biology becomes increasingly relevant. Mathematical models form an integral part of this research. This book presents: research on mathematical modeling of biological invasions, different forms of differential equations such as ordinary differential equations, reaction-diffusion models, and stochastic differential equations, detailed investigations as well as a thorough introduction to dynamical systems. Subjects range from theoretical to applied models on fascinating case studies, including: investigating how phenomena such as inducible defense or group defense can be incorporated into mathematical models, investigating how selective infections can influence the dispersal of invasive species, investigating how invasive mammals in New Zealand can be controlled to rescue the national bird, the Kiwi. The book is well suited not only for accomplished researchers but also for students with limited quantitative background interested in studying biological invasions.

Elliptic Partial Differential Equations

Author: Vitaly Volpert

Publisher: Springer

Published: 2014-05-10

Total Pages: 796

ISBN-13: 3034808135

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If we had to formulate in one sentence what this book is about, it might be "How partial differential equations can help to understand heat explosion, tumor growth or evolution of biological species". These and many other applications are described by reaction-diffusion equations. The theory of reaction-diffusion equations appeared in the first half of the last century. In the present time, it is widely used in population dynamics, chemical physics, biomedical modelling. The purpose of this book is to present the mathematical theory of reaction-diffusion equations in the context of their numerous applications. We will go from the general mathematical theory to specific equations and then to their applications. Existence, stability and bifurcations of solutions will be studied for bounded domains and in the case of travelling waves. The classical theory of reaction-diffusion equations and new topics such as nonlocal equations and multi-scale models in biology will be considered.

Biological Invasions

Author: M. Williamson

Publisher: Springer Science & Business Media

Published: 1996

Total Pages: 268

ISBN-13: 0412591901

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Some biological invasions have marked ecological and economic effects. But most fail, and most of those that succeed have small effects. This volume should be of interest to plant ecologists, plant conservationists, population biologists, agriculturalists

Biological Invasions: Theory and Practice

Author: Nanako Shigesada

Publisher: Oxford University Press, UK

Published: 1997-02-06

Total Pages: 222

ISBN-13: 0191589829

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This book deals with the ecological effect a species can have when it moves into an environment that it has not previously occupied (commonly referred to as an 'Invasion'). It is unique in presenting a clear and accessible introduction to a highly complex area - the modelling of biological invasions. The book presents the latest theories and models developed from studies into this crucial area. It includes data and examples from biological case studies showing how the models can be applied to the study of invasions, whether dealing with AIDS, the European rabbit, or prickly pear cactuses. - ;In nature, all organisms migrate or disperse to some extent, either by walking, swimming, flying, or being transported by wind or water. When a species succeeds in colonising an area that it has not previously inhabited, this is referred to as an `invasion'. Humans can precipitate biological invasions often spreading disease or pests by their travels around the world. Using the large amount of data that has been collected from studies worldwide, ranging from pest control to epidemiology, it has been possible to construct mathematical models that can predict which species will become an invader, what kind of habitat is susceptible to invasion by a particular species, and how fast an invasion will spread if it occurs. This book presents a clear and accessible introduction to this highly complex area. Included are data and examples from biological case studies showing how these models can be applied to the study of invasions, whether dealing with AIDS, the European rabbit, or prickly pear cactuses. -

Encyclopedia of Biological Invasions

Author: Daniel Simberloff

Publisher: Univ of California Press

Published: 2011-01-02

Total Pages: 792

ISBN-13: 0520264215

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"Addresses all aspects of this subject at a global level--including invasions by animals, plants, fungi, and bacteria--in succinct, alphabetically arranged articles. Featuring many cross-references, suggestions for further reading, illustrations, an appendix of the world's worst 100 invasive species, a glossary, and more..." -- From the publisher.

Invasion Dynamics

Author: Cang Hui

Publisher: Oxford University Press

Published: 2017

Total Pages: 337

ISBN-13: 0198745338

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An advanced textbook adopting a theoretical modeling approach to review and discuss the current range and distributions of alien species, their rates of spread, and their impact in human-dominated ecosystems.

Spatiotemporal Patterns in Models of Biological Invasion and Epidemic Spread

Author: Frank Mathias Hilker

Publisher:

Published: 2005

Total Pages: 0

ISBN-13: 9783832510503

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Biological invasions are a severe ecological problem threatening biodiversity and causing substantial economic damages. Mathematical models of spatiotemporal spread have proven to be powerful tools in identifying the underlying mechanisms, thus contributing to the understanding of the factors that determine invasion processes and to the assessment of possible control methods. In this thesis, classical models are extended to combine spatial spread, population growth, disease transmission and community interactions. Applications are exemplarily found in the circulation of the Feline Immunodeficiency Virus (FIV) - an HIV-similar lentivirus that induces AIDS in cat populations - and in viral infections in phytoplankton that forms the basis for all food chains and webs in the sea. The joint interplay of epidemics, predation and environmental stochasticity in invasion models is shown to generate rich and novel patterns of spatiotemporal spread such as the blocking and reversal of invasion fronts or the spatial `trapping' of infection as well as its noise-induced escape. The results of this thesis can explain real-world phenomena and have important implications for understanding and controlling invasion processes in ecosystems and epidemiology.

Biological Invasions, a Question of Nature and Society

Author: Robert Barbault

Publisher: Editions Quae

Published: 2011-01-17

Total Pages: 184

ISBN-13: 2759208729

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Invasions of introduced species cause varying degrees of harm on the ecosystems in question and it is up to society to deal with the consequences. How can we prevent biological invasions? How can we assess the risk they represent? What can be done to control current invasions? Aware of this problem, the Ministry of Ecology has requested a community of researchers from a variety of disciplines to decipher these questions using biological, sociological and economic approaches. Although the definitive response to the problems raised by invasive species in natural spaces is not provided here, undeniable progress in understanding mechanisms underlying these invasions can shed light on the decisions which have to be taken by environmental managers. Scientists, teachers and students will also find results and thought-processes in this book to supplement their knowledge.