Probabilistic Models of Population Evolution

Probabilistic Models of Population Evolution PDF

Author: Étienne Pardoux

Publisher: Springer

Published: 2016-06-17

Total Pages: 125

ISBN-13: 3319303287

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This expository book presents the mathematical description of evolutionary models of populations subject to interactions (e.g. competition) within the population. The author includes both models of finite populations, and limiting models as the size of the population tends to infinity. The size of the population is described as a random function of time and of the initial population (the ancestors at time 0). The genealogical tree of such a population is given. Most models imply that the population is bound to go extinct in finite time. It is explained when the interaction is strong enough so that the extinction time remains finite, when the ancestral population at time 0 goes to infinity. The material could be used for teaching stochastic processes, together with their applications. Étienne Pardoux is Professor at Aix-Marseille University, working in the field of Stochastic Analysis, stochastic partial differential equations, and probabilistic models in evolutionary biology and population genetics. He obtained his PhD in 1975 at University of Paris-Sud.

Probability Models for DNA Sequence Evolution

Probability Models for DNA Sequence Evolution PDF

Author: Rick Durrett

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 246

ISBN-13: 1475762852

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"What underlying forces are responsible for the observed patterns of variability, given a collection of DNA sequences?" In approaching this question a number of probability models are introduced and anyalyzed.Throughout the book, the theory is developed in close connection with data from more than 60 experimental studies that illustrate the use of these results.

A Biologist's Guide to Mathematical Modeling in Ecology and Evolution

A Biologist's Guide to Mathematical Modeling in Ecology and Evolution PDF

Author: Sarah P. Otto

Publisher: Princeton University Press

Published: 2011-09-19

Total Pages: 745

ISBN-13: 1400840910

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Thirty years ago, biologists could get by with a rudimentary grasp of mathematics and modeling. Not so today. In seeking to answer fundamental questions about how biological systems function and change over time, the modern biologist is as likely to rely on sophisticated mathematical and computer-based models as traditional fieldwork. In this book, Sarah Otto and Troy Day provide biology students with the tools necessary to both interpret models and to build their own. The book starts at an elementary level of mathematical modeling, assuming that the reader has had high school mathematics and first-year calculus. Otto and Day then gradually build in depth and complexity, from classic models in ecology and evolution to more intricate class-structured and probabilistic models. The authors provide primers with instructive exercises to introduce readers to the more advanced subjects of linear algebra and probability theory. Through examples, they describe how models have been used to understand such topics as the spread of HIV, chaos, the age structure of a country, speciation, and extinction. Ecologists and evolutionary biologists today need enough mathematical training to be able to assess the power and limits of biological models and to develop theories and models themselves. This innovative book will be an indispensable guide to the world of mathematical models for the next generation of biologists. A how-to guide for developing new mathematical models in biology Provides step-by-step recipes for constructing and analyzing models Interesting biological applications Explores classical models in ecology and evolution Questions at the end of every chapter Primers cover important mathematical topics Exercises with answers Appendixes summarize useful rules Labs and advanced material available

Population Dynamics: Algebraic And Probabilistic Approach

Population Dynamics: Algebraic And Probabilistic Approach PDF

Author: Utkir A Rozikov

Publisher: World Scientific

Published: 2020-04-22

Total Pages: 458

ISBN-13: 9811211248

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A population is a summation of all the organisms of the same group or species, which live in a particular geographical area, and have the capability of interbreeding. The main mathematical problem for a given population is to carefully examine the evolution (time dependent dynamics) of the population. The mathematical methods used in the study of this problem are based on probability theory, stochastic processes, dynamical systems, nonlinear differential and difference equations, and (non-)associative algebras.A state of a population is a distribution of probabilities of the different types of organisms in every generation. Type partition is called differentiation (for example, sex differentiation which defines a bisexual population). This book systematically describes the recently developed theory of (bisexual) population, and mainly contains results obtained since 2010.The book presents algebraic and probabilistic approaches in the theory of population dynamics. It also includes several dynamical systems of biological models such as dynamics generated by Markov processes of cubic stochastic matrices; dynamics of sex-linked population; dynamical systems generated by a gonosomal evolution operator; dynamical system and an evolution algebra of mosquito population; and ocean ecosystems.The main aim of this book is to facilitate the reader's in-depth understanding by giving a systematic review of the theory of population dynamics which has wide applications in biology, mathematics, medicine, and physics.

Stochastic Models for Structured Populations

Stochastic Models for Structured Populations PDF

Author: Sylvie Meleard

Publisher: Springer

Published: 2015-09-03

Total Pages: 107

ISBN-13: 3319217119

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In this contribution, several probabilistic tools to study population dynamics are developed. The focus is on scaling limits of qualitatively different stochastic individual based models and the long time behavior of some classes of limiting processes. Structured population dynamics are modeled by measure-valued processes describing the individual behaviors and taking into account the demographic and mutational parameters, and possible interactions between individuals. Many quantitative parameters appear in these models and several relevant normalizations are considered, leading to infinite-dimensional deterministic or stochastic large-population approximations. Biologically relevant questions are considered, such as extinction criteria, the effect of large birth events, the impact of environmental catastrophes, the mutation-selection trade-off, recovery criteria in parasite infections, genealogical properties of a sample of individuals. These notes originated from a lecture series on Structured Population Dynamics at Ecole polytechnique (France). Vincent Bansaye and Sylvie Méléard are Professors at Ecole Polytechnique (France). They are a specialists of branching processes and random particle systems in biology. Most of their research concerns the applications of probability to biodiversity, ecology and evolution.

Stability in Model Populations (MPB-31)

Stability in Model Populations (MPB-31) PDF

Author: Laurence D. Mueller

Publisher: Princeton University Press

Published: 2020-03-31

Total Pages: 334

ISBN-13: 0691209944

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Throughout the twentieth century, biologists investigated the mechanisms that stabilize biological populations, populations which--if unchecked by such agencies as competition and predation--should grow geometrically. How is order in nature maintained in the face of the seemingly disorderly struggle for existence? In this book, Laurence Mueller and Amitabh Joshi examine current theories of population stability and show how recent laboratory research on model populations--particularly blowflies, Tribolium, and Drosophila--contributes to our understanding of population dynamics and the evolution of stability. The authors review the general theory of population stability and critically analyze techniques for inferring whether a given population is in balance or not. They then show how rigorous empirical research can reveal both the proximal causes of stability (how populations are regulated and maintained at an equilibrium, including the relative roles of biotic and abiotic factors) and its ultimate, mostly evolutionary causes. In the process, they describe experimental studies on model systems that address the effects of age-structure, inbreeding, resource levels, and population structure on the stability and persistence of populations. The discussion incorporates the authors' own findings on the evolution of population stability in Drosophila. They go on to relate laboratory work to studies of animals in the wild and to develop a general framework for relating the life history and ecology of a species to its population dynamics. This accessible, finely written illustration of how carefully designed experiments can improve theory will have tremendous value for all ecologists and evolutionary biologists.

Some Mathematical Models from Population Genetics

Some Mathematical Models from Population Genetics PDF

Author: Alison Etheridge

Publisher: Springer Science & Business Media

Published: 2011-01-07

Total Pages: 129

ISBN-13: 3642166318

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This work reflects sixteen hours of lectures delivered by the author at the 2009 St Flour summer school in probability. It provides a rapid introduction to a range of mathematical models that have their origins in theoretical population genetics. The models fall into two classes: forwards in time models for the evolution of frequencies of different genetic types in a population; and backwards in time (coalescent) models that trace out the genealogical relationships between individuals in a sample from the population. Some, like the classical Wright-Fisher model, date right back to the origins of the subject. Others, like the multiple merger coalescents or the spatial Lambda-Fleming-Viot process are much more recent. All share a rich mathematical structure. Biological terms are explained, the models are carefully motivated and tools for their study are presented systematically.

Population Dynamics

Population Dynamics PDF

Author: Bertram G. Jr. Murray

Publisher: Elsevier

Published: 2013-07-22

Total Pages: 223

ISBN-13: 0323159850

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Population Dynamics: Alternative Models provides a theoretical framework of population dynamics. This book contains seven chapters that discuss the controversies surrounding discussions on the explicit view of the subject. Chapters 1 and 2 present a general introduction to the terminology, the mathematical background, and the philosophical approach that lie behind the theoretical development. Chapter 3 contains a series of models accounting for variations in population growth rates, sizes, and fluctuations, while Chapter 4 examines a model accounting for the evolution of life history patterns. A more detailed examination of the effects of predation on prey populations, especially with respect to determining a prey population's maximum sustainable yield, is explored in Chapter 5. Chapter 6 highlights the interspecific competition theory in terms of the population dynamics models presented in a previous chapter. Chapter 7 summarizes the developments in the population dynamics research studies. This work will be of great value to ecologists, biologists, and population dynamics researchers.

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