Matrix Models For Population Disease And Evolutionary Dynamics

This book offers an introduction to the use of matrix theory and linear algebra in modeling the dynamics of biological populations. Matrix algebra has been used in population biology since the 1940s and continues to play a major role in theoretical and applied dynamics for populations structured by age, body size or weight, disease states, physiological and behavioral characteristics, life cycle stages, or any of many other possible classification schemes. With a focus on matrix models, the book requires only first courses in multivariable calculus and matrix theory or linear algebra as prerequisites. The reader will learn the basics of modeling methodology (i.e., how to set up a matrix model from biological underpinnings) and the fundamentals of the analysis of discrete time dynamical systems (equilibria, stability, bifurcations, etc.). A recurrent theme in all chapters concerns the problem of extinction versus survival of a population. In addition to numerous examples that illustrate these fundamentals, several applications appear at the end of each chapter that illustrate the full cycle of model setup, mathematical analysis, and interpretation. The author has used the material over many decades in a variety of teaching and mentoring settings, including special topics courses and seminars in mathematical modeling, mathematical biology, and dynamical systems.

Chaos in Ecology is a convincing demonstration of chaos in a biological population. The book synthesizes an ecologically focused interdisciplinary blend of non-linear dynamics theory, statistics, and experimentation yielding results of uncommon clarity and rigor. Topics include fundamental issues that are of general and widespread importance to population biology and ecology. Detailed descriptions are included of the mathematical, statistical, and experimental steps they used to explore nonlinear dynamics in ecology. Beginning with a brief overview of chaos theory and its implications for ecology. The book continues by deriving and rigorously testing a mathematical model that is closely wedded to biological mechanisms of their research organism. Therefrom were generated a variety of predictions that are fundamental to chaos theory and experiments were designed and analyzed to test those predictions. Discussion of patterns in chaos and how they can be investigated using real data follows and book ends with a discussion of the salient lessons learned from this research program Book jacket.

Interest in the temporal fluctuations of biological populations can be traced to the dawn of civilization. How can mathematics be used to gain an understanding of population dynamics? This monograph introduces the theory of structured population dynamics and its applications, focusing on the asymptotic dynamics of deterministic models. This theory bridges the gap between the characteristics of individual organisms in a population and the dynamics of the total population as a whole. In this monograph, many applications that illustrate both the theory and a wide variety of biological issues are given, along with an interdisciplinary case study that illustrates the connection of models with the data and the experimental documentation of model predictions. The author also discusses the use of discrete and continuous models and presents a general modeling theory for structured population dynamics. Cushing begins with an obvious point: individuals in biological populations differ with regard to their physical and behavioral characteristics and therefore in the way they interact with their environment. Studying this point effectively requires the use of structured models. Specific examples cited throughout support the valuable use of structured models. Included among these are important applications chosen to illustrate both the mathematical theories and biological problems that have received attention in recent literature.

This textbook offers a rigorous mathematical introduction to cellular automata (CA). Numerous colorful graphics illustrate the many intriguing phenomena, inviting undergraduates to step into the rich field of symbolic dynamics. Beginning with a brief history, the first half of the book establishes the mathematical foundations of cellular automata. After recapping the essentials from advanced calculus, the chapters that follow introduce symbolic spaces, equicontinuity, and attractors. More advanced topics include the Garden of Eden theorem and Conway's Game of Life, and a chapter on stochastic CA showcases a model of virus spread. Exercises and labs end each chapter, covering a range of applications, both mathematical and physical. Designed for undergraduates studying mathematics and related areas, the text provides ample opportunities for end-of-semester projects or further study. Computer use for the labs is largely optional, providing flexibility for different preferences and resources. Knowledge of advanced calculus and linear algebra is essential, while a course in real analysis would be ideal.

This book gives a unifying framework for estimating the abundance of open populations: populations subject to births, deaths and movement, given imperfect measurements or samples of the populations. The focus is primarily on populations of vertebrates for which dynamics are typically modelled within the framework of an annual cycle, and for which stochastic variability in the demographic processes is usually modest. Discrete-time models are developed in which animals can be assigned to discrete states such as age class, gender, maturity, population (within a metapopulation), or species (for multi-species models). The book goes well beyond estimation of abundance, allowing inference on underlying population processes such as birth or recruitment, survival and movement. This requires the formulation and fitting of population dynamics models. The resulting fitted models yield both estimates of abundance and estimates of parameters characterizing the underlying processes.

Updated to include two new chapters, a modified Part II structure, more recent empirical examples, and online spreadsheet simulations.

Mankind now faces even more challenging environment- and health-related problems than ever before. Readily available transportation systems facilitate the swift spread of diseases as large populations migrate from one part of the world to another. Studies on the spread of the communicable diseases are very important. This book, Mathematical Population Dynamics and Epidemiology in Temporal and Spatio-Temporal Domains, provides a useful experimental tool for making practical predictions, building and testing theories, answering specific questions, determining sensitivities of the parameters, forming control strategies, and much more. This volume focuses on the study of population dynamics with special emphasis on the migration of populations and the spreading of epidemics among human and animal populations. It also provides the background needed to interpret, construct, and analyze a wide variety of mathematical models. Most of the techniques presented in the book can be readily applied to model other phenomena, in biology as well as in other disciplines.

This open access book shows how to use sensitivity analysis in demography. It presents new methods for individuals, cohorts, and populations, with applications to humans, other animals, and plants. The analyses are based on matrix formulations of age-classified, stage-classified, and multistate population models. Methods are presented for linear and nonlinear, deterministic and stochastic, and time-invariant and time-varying cases. Readers will discover results on the sensitivity of statistics of longevity, life disparity, occupancy times, the net reproductive rate, and statistics of Markov chain models in demography. They will also see applications of sensitivity analysis to population growth rates, stable population structures, reproductive value, equilibria under immigration and nonlinearity, and population cycles. Individual stochasticity is a theme throughout, with a focus that goes beyond expected values to include variances in demographic outcomes. The calculations are easily and accurately implemented in matrix-oriented programming languages such as Matlab or R. Sensitivity analysis will help readers create models to predict the effect of future changes, to evaluate policy effects, and to identify possible evolutionary responses to the environment. Complete with many examples of the application, the book will be of interest to researchers and graduate students in human demography and population biology. The material will also appeal to those in mathematical biology and applied mathematics.